cauchy-euler equations - university of southern mississippi€¦ · cauchy-euler equations bernd...
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![Page 1: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/1.jpg)
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Overview An Example Double Check Discussion
Cauchy-Euler Equations
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 2: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/2.jpg)
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Overview An Example Double Check Discussion
Definition and Solution Method
1. A second order Cauchy-Euler equation is of the form
a2x2 d2ydx2 +a1x
dydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.2. To solve a homogeneous Cauchy-Euler equation we set
y = xr and solve for r.3. The idea is similar to that for homogeneous linear
differential equations with constant coefficients. We willuse this similarity in the final discussion.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 3: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/3.jpg)
logo1
Overview An Example Double Check Discussion
Definition and Solution Method1. A second order Cauchy-Euler equation is of the form
a2x2 d2ydx2 +a1x
dydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.2. To solve a homogeneous Cauchy-Euler equation we set
y = xr and solve for r.3. The idea is similar to that for homogeneous linear
differential equations with constant coefficients. We willuse this similarity in the final discussion.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 4: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/4.jpg)
logo1
Overview An Example Double Check Discussion
Definition and Solution Method1. A second order Cauchy-Euler equation is of the form
a2x2 d2ydx2 +a1x
dydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.
2. To solve a homogeneous Cauchy-Euler equation we sety = xr and solve for r.
3. The idea is similar to that for homogeneous lineardifferential equations with constant coefficients. We willuse this similarity in the final discussion.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 5: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/5.jpg)
logo1
Overview An Example Double Check Discussion
Definition and Solution Method1. A second order Cauchy-Euler equation is of the form
a2x2 d2ydx2 +a1x
dydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.2. To solve a homogeneous Cauchy-Euler equation we set
y = xr and solve for r.
3. The idea is similar to that for homogeneous lineardifferential equations with constant coefficients. We willuse this similarity in the final discussion.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 6: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/6.jpg)
logo1
Overview An Example Double Check Discussion
Definition and Solution Method1. A second order Cauchy-Euler equation is of the form
a2x2 d2ydx2 +a1x
dydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.2. To solve a homogeneous Cauchy-Euler equation we set
y = xr and solve for r.3. The idea is similar to that for homogeneous linear
differential equations with constant coefficients.
We willuse this similarity in the final discussion.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 7: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/7.jpg)
logo1
Overview An Example Double Check Discussion
Definition and Solution Method1. A second order Cauchy-Euler equation is of the form
a2x2 d2ydx2 +a1x
dydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.2. To solve a homogeneous Cauchy-Euler equation we set
y = xr and solve for r.3. The idea is similar to that for homogeneous linear
differential equations with constant coefficients. We willuse this similarity in the final discussion.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 8: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/8.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 9: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/9.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 0
2x2r(r−1)xr−2 + xrxr−1− xr = 02r(r−1)xr + rxr − xr = 0
2r(r−1)+ r−1 = 02r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 10: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/10.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2
+ xrxr−1− xr = 02r(r−1)xr + rxr − xr = 0
2r(r−1)+ r−1 = 02r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 11: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/11.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1
− xr = 02r(r−1)xr + rxr − xr = 0
2r(r−1)+ r−1 = 02r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 12: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/12.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr
= 02r(r−1)xr + rxr − xr = 0
2r(r−1)+ r−1 = 02r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 13: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/13.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 14: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/14.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr
+ rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 15: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/15.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr
− xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 16: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/16.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr
= 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 17: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/17.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 0
2r(r−1)+ r−1 = 02r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 18: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/18.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 19: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/19.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 20: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/20.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2
=1±3
4= 1,−1
2
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 21: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/21.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 22: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/22.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 23: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/23.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12
= c1x+ c21√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 24: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/24.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
2x2y′′+ xy′− y = 02x2r(r−1)xr−2 + xrxr−1− xr = 0
2r(r−1)xr + rxr − xr = 02r(r−1)+ r−1 = 0
2r2− r−1 = 0
r1,2 =−(−1)±
√(−1)2−4 ·2 · (−1)
2 ·2=
1±34
= 1,−12
y(x) = c1x1 + c2x−12 = c1x+ c2
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 25: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/25.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2, c2 =−23, c1 = 1− c2 =
53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 26: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/26.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2, c2 =−23, c1 = 1− c2 =
53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 27: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/27.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2, c2 =−23, c1 = 1− c2 =
53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 28: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/28.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2, c2 =−23, c1 = 1− c2 =
53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 29: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/29.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2, c2 =−23, c1 = 1− c2 =
53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 30: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/30.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2
, c2 =−23, c1 = 1− c2 =
53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 31: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/31.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2, c2 =−23
, c1 = 1− c2 =53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 32: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/32.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2, c2 =−23, c1 = 1− c2
=53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 33: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/33.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2, c2 =−23, c1 = 1− c2 =
53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 34: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/34.jpg)
logo1
Overview An Example Double Check Discussion
Solve the Initial Value Problem2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) = c1x+ c21√x
y′(x) = c1−12
c21
x32
1 = y(1) = c1 + c2
2 = y′(1) = c1−12
c2
−1 =32
c2, c2 =−23, c1 = 1− c2 =
53
y(x) =53
x− 23
1√x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 35: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/35.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 36: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/36.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 37: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/37.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1
√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 38: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/38.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 39: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/39.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 40: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/40.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2
√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 41: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/41.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 42: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/42.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 43: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/43.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)
+ x(
53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 44: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/44.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)
−(
53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 45: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/45.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)
?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 46: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/46.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 47: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/47.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 48: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/48.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12
− 53
x+23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 49: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/49.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 50: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/50.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 51: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/51.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x
(53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 52: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/52.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53
− 53
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 53: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/53.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)
+1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 54: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/54.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 55: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/55.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)
+13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 56: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/56.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 57: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/57.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)
?= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 58: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/58.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 59: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/59.jpg)
logo1
Overview An Example Double Check Discussion
Does y(x) =53
x− 23
1√x
Really Solve the Initial Value Problem
2x2y′′+ xy′− y = 0, y(1) = 1, y′(1) = 2
y(x) =53
x− 23
1√x
, y(1) = 1√
y′(x) =53
+13
1
x32
, y′(1) = 2√
y′′(x) =−12
1
x52
2x2(−1
21
x52
)+ x
(53
+13
1
x32
)−
(53
x− 23
1√x
)?= 0
− 1
x12
+53
x+13
1
x12− 5
3x+
23
1
x12
?= 0
x(
53− 5
3
)+
1
x12
((−1)+
13
+23
)?= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 60: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/60.jpg)
logo1
Overview An Example Double Check Discussion
General Solution Method
1. Inhomogeneous Cauchy-Euler equations are solved withVariation of Parameters.
2. A Cauchy-Euler equation is of the form
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.3. The substitution t = ln(x) turns the Cauchy-Euler equation
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x)
for x > 0 into a linear differential equation with constantcoefficients.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 61: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/61.jpg)
logo1
Overview An Example Double Check Discussion
General Solution Method1. Inhomogeneous Cauchy-Euler equations are solved with
Variation of Parameters.
2. A Cauchy-Euler equation is of the form
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.3. The substitution t = ln(x) turns the Cauchy-Euler equation
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x)
for x > 0 into a linear differential equation with constantcoefficients.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 62: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/62.jpg)
logo1
Overview An Example Double Check Discussion
General Solution Method1. Inhomogeneous Cauchy-Euler equations are solved with
Variation of Parameters.2. A Cauchy-Euler equation is of the form
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.3. The substitution t = ln(x) turns the Cauchy-Euler equation
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x)
for x > 0 into a linear differential equation with constantcoefficients.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 63: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/63.jpg)
logo1
Overview An Example Double Check Discussion
General Solution Method1. Inhomogeneous Cauchy-Euler equations are solved with
Variation of Parameters.2. A Cauchy-Euler equation is of the form
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.
3. The substitution t = ln(x) turns the Cauchy-Euler equation
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x)
for x > 0 into a linear differential equation with constantcoefficients.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 64: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/64.jpg)
logo1
Overview An Example Double Check Discussion
General Solution Method1. Inhomogeneous Cauchy-Euler equations are solved with
Variation of Parameters.2. A Cauchy-Euler equation is of the form
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x).
If g(x) = 0, then the equation is called homogeneous.3. The substitution t = ln(x) turns the Cauchy-Euler equation
anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = g(x)
for x > 0 into a linear differential equation with constantcoefficients.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 65: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/65.jpg)
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Overview An Example Double Check Discussion
General Solution Method
Solving anxn dnydxn +an−1xn−1 dn−1y
dxn−1 + · · ·+a1xdydx
+a0y = 0.
1. Substitute y = xr into the equation.2. Cancel xr to obtain an equation p(r) = 0 for r.3. Find the solutions r1, . . . ,rm of p(r) = 0.4. If rk is real then y(x) = xrk solves the differential equation.5. If rk = ak + ibk is complex then y(x) = xak cos
(bk ln(x)
)and y(x) = xak sin
(bk ln(x)
)solve the differential equation.
6. If (r− rk)j is a factor of p(r), then xrk , ln(x)xrk , . . . ,(ln(x)
)j−1xrk solve the differential equation. (If rk iscomplex we need to multiply the solutions from 5 with thepower of the logarithm.)
7. The general solution is a linear combination of thesolutions above with generic coefficients c1, . . . ,cn.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 66: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/66.jpg)
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Overview An Example Double Check Discussion
General Solution MethodSolving anxn dny
dxn +an−1xn−1 dn−1ydxn−1 + · · ·+a1x
dydx
+a0y = 0.
1. Substitute y = xr into the equation.2. Cancel xr to obtain an equation p(r) = 0 for r.3. Find the solutions r1, . . . ,rm of p(r) = 0.4. If rk is real then y(x) = xrk solves the differential equation.5. If rk = ak + ibk is complex then y(x) = xak cos
(bk ln(x)
)and y(x) = xak sin
(bk ln(x)
)solve the differential equation.
6. If (r− rk)j is a factor of p(r), then xrk , ln(x)xrk , . . . ,(ln(x)
)j−1xrk solve the differential equation. (If rk iscomplex we need to multiply the solutions from 5 with thepower of the logarithm.)
7. The general solution is a linear combination of thesolutions above with generic coefficients c1, . . . ,cn.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 67: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/67.jpg)
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Overview An Example Double Check Discussion
General Solution MethodSolving anxn dny
dxn +an−1xn−1 dn−1ydxn−1 + · · ·+a1x
dydx
+a0y = 0.
1. Substitute y = xr into the equation.
2. Cancel xr to obtain an equation p(r) = 0 for r.3. Find the solutions r1, . . . ,rm of p(r) = 0.4. If rk is real then y(x) = xrk solves the differential equation.5. If rk = ak + ibk is complex then y(x) = xak cos
(bk ln(x)
)and y(x) = xak sin
(bk ln(x)
)solve the differential equation.
6. If (r− rk)j is a factor of p(r), then xrk , ln(x)xrk , . . . ,(ln(x)
)j−1xrk solve the differential equation. (If rk iscomplex we need to multiply the solutions from 5 with thepower of the logarithm.)
7. The general solution is a linear combination of thesolutions above with generic coefficients c1, . . . ,cn.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 68: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/68.jpg)
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Overview An Example Double Check Discussion
General Solution MethodSolving anxn dny
dxn +an−1xn−1 dn−1ydxn−1 + · · ·+a1x
dydx
+a0y = 0.
1. Substitute y = xr into the equation.2. Cancel xr to obtain an equation p(r) = 0 for r.
3. Find the solutions r1, . . . ,rm of p(r) = 0.4. If rk is real then y(x) = xrk solves the differential equation.5. If rk = ak + ibk is complex then y(x) = xak cos
(bk ln(x)
)and y(x) = xak sin
(bk ln(x)
)solve the differential equation.
6. If (r− rk)j is a factor of p(r), then xrk , ln(x)xrk , . . . ,(ln(x)
)j−1xrk solve the differential equation. (If rk iscomplex we need to multiply the solutions from 5 with thepower of the logarithm.)
7. The general solution is a linear combination of thesolutions above with generic coefficients c1, . . . ,cn.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 69: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/69.jpg)
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Overview An Example Double Check Discussion
General Solution MethodSolving anxn dny
dxn +an−1xn−1 dn−1ydxn−1 + · · ·+a1x
dydx
+a0y = 0.
1. Substitute y = xr into the equation.2. Cancel xr to obtain an equation p(r) = 0 for r.3. Find the solutions r1, . . . ,rm of p(r) = 0.
4. If rk is real then y(x) = xrk solves the differential equation.5. If rk = ak + ibk is complex then y(x) = xak cos
(bk ln(x)
)and y(x) = xak sin
(bk ln(x)
)solve the differential equation.
6. If (r− rk)j is a factor of p(r), then xrk , ln(x)xrk , . . . ,(ln(x)
)j−1xrk solve the differential equation. (If rk iscomplex we need to multiply the solutions from 5 with thepower of the logarithm.)
7. The general solution is a linear combination of thesolutions above with generic coefficients c1, . . . ,cn.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 70: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/70.jpg)
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Overview An Example Double Check Discussion
General Solution MethodSolving anxn dny
dxn +an−1xn−1 dn−1ydxn−1 + · · ·+a1x
dydx
+a0y = 0.
1. Substitute y = xr into the equation.2. Cancel xr to obtain an equation p(r) = 0 for r.3. Find the solutions r1, . . . ,rm of p(r) = 0.4. If rk is real then y(x) = xrk solves the differential equation.
5. If rk = ak + ibk is complex then y(x) = xak cos(bk ln(x)
)and y(x) = xak sin
(bk ln(x)
)solve the differential equation.
6. If (r− rk)j is a factor of p(r), then xrk , ln(x)xrk , . . . ,(ln(x)
)j−1xrk solve the differential equation. (If rk iscomplex we need to multiply the solutions from 5 with thepower of the logarithm.)
7. The general solution is a linear combination of thesolutions above with generic coefficients c1, . . . ,cn.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 71: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/71.jpg)
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Overview An Example Double Check Discussion
General Solution MethodSolving anxn dny
dxn +an−1xn−1 dn−1ydxn−1 + · · ·+a1x
dydx
+a0y = 0.
1. Substitute y = xr into the equation.2. Cancel xr to obtain an equation p(r) = 0 for r.3. Find the solutions r1, . . . ,rm of p(r) = 0.4. If rk is real then y(x) = xrk solves the differential equation.5. If rk = ak + ibk is complex then y(x) = xak cos
(bk ln(x)
)and y(x) = xak sin
(bk ln(x)
)solve the differential equation.
6. If (r− rk)j is a factor of p(r), then xrk , ln(x)xrk , . . . ,(ln(x)
)j−1xrk solve the differential equation. (If rk iscomplex we need to multiply the solutions from 5 with thepower of the logarithm.)
7. The general solution is a linear combination of thesolutions above with generic coefficients c1, . . . ,cn.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 72: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/72.jpg)
logo1
Overview An Example Double Check Discussion
General Solution MethodSolving anxn dny
dxn +an−1xn−1 dn−1ydxn−1 + · · ·+a1x
dydx
+a0y = 0.
1. Substitute y = xr into the equation.2. Cancel xr to obtain an equation p(r) = 0 for r.3. Find the solutions r1, . . . ,rm of p(r) = 0.4. If rk is real then y(x) = xrk solves the differential equation.5. If rk = ak + ibk is complex then y(x) = xak cos
(bk ln(x)
)and y(x) = xak sin
(bk ln(x)
)solve the differential equation.
6. If (r− rk)j is a factor of p(r), then xrk , ln(x)xrk , . . . ,(ln(x)
)j−1xrk solve the differential equation. (If rk iscomplex we need to multiply the solutions from 5 with thepower of the logarithm.)
7. The general solution is a linear combination of thesolutions above with generic coefficients c1, . . . ,cn.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations
![Page 73: Cauchy-Euler Equations - University of Southern Mississippi€¦ · Cauchy-Euler Equations Bernd Schroder¨ Bernd Schroder¨ Louisiana Tech University, College of Engineering and](https://reader036.vdocuments.mx/reader036/viewer/2022081514/5edf6797ad6a402d666ac081/html5/thumbnails/73.jpg)
logo1
Overview An Example Double Check Discussion
General Solution MethodSolving anxn dny
dxn +an−1xn−1 dn−1ydxn−1 + · · ·+a1x
dydx
+a0y = 0.
1. Substitute y = xr into the equation.2. Cancel xr to obtain an equation p(r) = 0 for r.3. Find the solutions r1, . . . ,rm of p(r) = 0.4. If rk is real then y(x) = xrk solves the differential equation.5. If rk = ak + ibk is complex then y(x) = xak cos
(bk ln(x)
)and y(x) = xak sin
(bk ln(x)
)solve the differential equation.
6. If (r− rk)j is a factor of p(r), then xrk , ln(x)xrk , . . . ,(ln(x)
)j−1xrk solve the differential equation. (If rk iscomplex we need to multiply the solutions from 5 with thepower of the logarithm.)
7. The general solution is a linear combination of thesolutions above with generic coefficients c1, . . . ,cn.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Cauchy-Euler Equations