linear first order differential equations · 2008-11-06 · recognizing linear first order...
TRANSCRIPT
logo1
Overview An Example Double Check
Linear First Order Differential Equations
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.
3.1 Compute the integrating factor µ(x) = e∫
p(x) dx,3.2 Multiply the equation by the integrating factor,
note that the left side e∫
p(x) dxy′ + e∫
p(x) dxp(x)yis the derivative of the product e
∫p(x) dxy,
3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,
note that the left side e∫
p(x) dxy′ + e∫
p(x) dxp(x)yis the derivative of the product e
∫p(x) dxy,
3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,
3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
What are Linear First Order DifferentialEquations?
1. A linear first order differential equation is of the formy′ +p(x)y = q(x).
2. Recognizing linear first order differential equationsrequires some pattern recognition.
3. To solve a linear first order differential equation, we turnthe left side into the derivative of a product.3.1 Compute the integrating factor µ(x) = e
∫p(x) dx,
3.2 Multiply the equation by the integrating factor,note that the left side e
∫p(x) dxy′ + e
∫p(x) dxp(x)y
is the derivative of the product e∫
p(x) dxy,3.3 Integrate both sides, solve for y.
That’s it.Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
Integrating factor:∫p(x) dx =
∫ cos(x)sin(x)
dx u := sin(x),dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =
∫ cos(x)sin(x)
dx u := sin(x),dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx
u := sin(x),dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
dx =du
cos(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|+ c
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|+ c
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|+ c
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|
= ln∣∣sin(x)
∣∣µ(x) = e
∫p(x) dx = eln
∣∣sin(x)∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) = e∫
p(x) dx = eln∣∣sin(x)
∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) =
e∫
p(x) dx = eln∣∣sin(x)
∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) = e∫
p(x) dx
= eln∣∣sin(x)
∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) = e∫
p(x) dx = eln∣∣sin(x)
∣∣
= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.Integrating factor:∫
p(x) dx =∫ cos(x)
sin(x)dx u := sin(x),
dudx
= cos(x),
=∫ cos(x)
udu
cos(x)dx =
ducos(x)
=∫ 1
udu
= ln |u|= ln
∣∣sin(x)∣∣
µ(x) = e∫
p(x) dx = eln∣∣sin(x)
∣∣= sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y′ +cos(x)sin(x)
y = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.y′ +
cos(x)sin(x)
y = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.µ(x)
(y′ +
cos(x)sin(x)
y)
= 1 ·µ(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)
ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.sin(x)
(y′ +
cos(x)sin(x)
y)
= 1 · sin(x)
sin(x)y′ + cos(x)y = sin(x)ddx
(sin(x)y
)= sin(x)
sin(x)y = −cos(x)+ c
y = −cos(x)sin(x)
+c
sin(x)
= −cot(x)+c
sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1)
= −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)
csin(1)
= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1)
= cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+c
sin(x)
0 = y(1) = −cot(1)+c
sin(1)c
sin(1)= cot(1)
c = cot(1)sin(1) = cos(1)
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Solve the Initial Value Problem y′+cos(x)sin(x)
y = 1,
y(1) = 0.
y(x) = −cot(x)+cos(1)sin(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y′ +cos(x)sin(x)
y ?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y′ +cos(x)sin(x)
y ?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)
?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
?= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
ddx
(−cot(x)+
cos(1)sin(x)
)+
cos(x)sin(x)
(−cot(x)+
cos(1)sin(x)
)?= 1
1sin2(x)
− cos(1)cos(x)sin2(x)
− cos2(x)sin2(x)
+cos(1)cos(x)
sin2(x)?= 1
1− cos2(x)sin2(x)
?= 1
sin2(x)sin2(x)
= 1√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) =
− cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0
√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations
logo1
Overview An Example Double Check
Does y(x) = −cot(x)+cos(1)sin(x)
Really Solve the
Initial Value Problem y′+cos(x)sin(x)
y = 1, y(1) = 0?
y(1) = − cot(1)+cos(1)sin(1)
= 0√
Yes, it does
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Linear First Order Differential Equations