section 2.1 introduction: second-order linear equations

Section 2.1 Introduction:

Second-Order Linear Equations

Second Order Differential Equations: F(x, y, y′, y″) = 0  

 2nd Order Linear Differential Equations: A(x) y″ + B(x) y′ + C(x) y = f (x)

often these will be in the form y″ + P(x) y′ + Q(x) y = f (x)

nth Order Linear Differential Equations

Homogeneous Linear Differential Equations: A(x) y″ + B(x) y′ + C(x) y = 0

Solutions to homogeneous linear Differential Equations form vector spaces!

TheoremLet y1 and y2 be solutions to y″ + P(x) y′ + Q(x) y = 0. Then functions of the form f (x) = c1 y1 + c2 y2 (with c1 and c2 being any constants) are also solutions.

General Solutions

Initial Conditions

TheoremSuppose the P(x) and Q(x) are continuous on an interval J containing xo. Then the initial value problem y″ + P(x) y′ + Q(x) y = 0, y′(xo) = a , y(xo) = b has a unique solution on J.

Ex. 1 Solve y″ + y = 0, 4 42 5 2,

2 2y y

Suppose we've found two linearly independent solutions, y1 and y2, to the initial value problem y″ + P(x) y′ + Q(x) y = 0, y′(xo) = a, y(xo) = b. How do we find the general solution to this differential equation?

Recall:

Two functions f (x) and g(x) are said to be linearly independent if one function cannot be written as a multiple of the other (or equivalently, f (x) and g(x) are linearly independent if the equation c1 f (x) + c2 g(x) = 0 has only a solution of c1 = c2 = 0).

Ex. 2 Show that y1 = sin(x) and y2 = 3sin(x) are both solutions to y″ + y = 0,

but we cannot use these to find our particular solution. 4

2 ,2

y 45 2

2y

Definition:The Wronskian of f (x) and g(x) is defined as: ( , )

f gW f g

f g

Ex. 3 Calculate the Wronskian of f (x) = sin(x) and g(x) = x2.

Theorem:Suppose y1 and y2 are solutions of y″ + P(x) y′ + Q(x) y = 0 on an interval J in which P(x) and Q(x) are continuous.

(a) If y1 and y2 are linearly dependent then W(f, g) = 0 for all x on the interval J.

(b) If y1 and y2 are linearly independent then W(f, g) ≠ 0 for any x on the interval J.

Ex. 4 (a) What does this theorem imply regarding the solutions y1 = sin(x) and y2 = cos(x) to the differential equation y″ + y = 0?

Ex. 4 (b) What does this theorem imply regarding the solutions y1 = sin(x) and y2 = 3sin(x) to the differential equation y″ + y = 0?

Theorem:Suppose y1 and y2 are linearly independent solutions of y″ + P(x) y′ + Q(x) y = 0 where P(x) and Q(x) are continuous on an interval J. If g(x) is any solution to this differential equation then g(x) = c1 y1 + c2 y2 for some choices of constants c1 and c2.

Big Important Question:

How do we determine two linearly independent solutions, y1 and y2?We shall examine this question for the differential equation ay″ + by′ + cy = 0 (where a, b, and c are constants).

Claim: y = erx is a solution (for some constant r) to ay″ + by′ + cy = 0. We just have to figure out what value of r will work.

Ex. 5 Solve y″ – 5y′ + 6y = 0, y(0) = 1, y′(0) = –1.

Theorem:

If the roots of the characteristic equation ar2 + br + c = 0 are r1 and r2 with

r1 ≠ r2 then y1 = _________ and y2 = _________ are two linearly independent

solutions to the differential equation ay″ + by′ + cy = 0.

Thus, y = _________________ is the general solution to ay″ + by′ + cy = 0.

What if there is a repeated root of the characteristic equation?   

Ex. 6 Solve y″ – 8y′ + 16y = 0, y(0) = 5, y′(0) = 21.

Theorem:

If r1 is a repeated root of the characteristic equation ar2 + br + c = 0 then

y1 = _________ and y2 = _________ are two linearly independent solutions

to the differential equation ay″ + by′ + cy = 0. Thus,

y = _________________ is the general solution to ay″ + by′ + cy = 0.

Ex. 6 Solve y″ – 8y′ + 16y = 0, y(0) = 5, y′(0) = 21.

Why does this theorem work? Let's prove that it produces the correct general solution for this past problem. (Introduction to the linear operator D.)

Why does this theorem work? Let's prove that it produces the correct general solution for this past problem. (Introduction to the linear operator D.)

Why does this theorem work? Let's prove that it produces the correct general solution for this past problem. (Introduction to the linear operator D.)

Why does y″ + y = 0 have solutions of y1 = sin(x) and y2 = cos(x)????Our previous theorems couldn't have possibly generated these solutions.

Ex. 7 Use our past two theorems to solve y″ + y = 0.

Recall that ez =

2 3 4 51 1 1 1 1 111! 2! 3! 4! 5! !

nz z z z z zn

Recall that ez = cos(z) =

2 3 4 51 1 1 1 1 111! 2! 3! 4! 5! !

nz z z z z zn

2 4 211 112! 4! 2 !

nnz z z

n

Recall that ez = cos(z) =

sin(z) =

2 3 4 51 1 1 1 1 111! 2! 3! 4! 5! !

nz z z z z zn

2 4 211 112! 4! 2 !

nnz z z

n

3 5 2 111 1 11! 3! 5! 2 1 !

nnz z z z

n

Recall that ez = cos(z) =

sin(z) =

2 3 4 51 1 1 1 1 111! 2! 3! 4! 5! !

nz z z z z zn

2 4 211 112! 4! 2 !

nnz z z

n

3 5 2 111 1 11! 3! 5! 2 1 !

nnz z z z

n

Ex. 8 Solve y″ – 6y′ + 13y = 0

Theorem:If the roots of the characteristic equation ar2 + br + c = 0 are A ± Bi then y1 = eAx sin(Bx) and y2 = eAx cos(Bx) are two linearly independent solutions to the differential equation ay″ + by′ + y = 0. Thus, y = c1 eAx sin(Bx) + c2 eAx cos(Bx) is the general solution to ay″ + by′ + y = 0.

Section 2.2 General Solutions of Linear Equations

Higher order linear differential equations:

y(n) + pn–1(x) y(n–1) + pn–2(x) y(n–2) + · · · · · + p2(x) y″ + p1(x) y′ + p0(x) y = f (x)

If y1, y2, . . . , yn are linearly independent solutions to the homogeneous diff. eq.

y(n) + pn–1(x) y(n–1) + pn–2(x) y(n–2) + · · · · · + p1(x) y′ + p0(x) y = 0,

then the general solution to y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = 0 is

y = c1y1 + c2y2 + · · · + cnyn.

To prove this we needed the following definitions and theorems:

Theorem: If y1, y2, . . . , yn are solutions to y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = 0 then so is y = c1y1 + c2y2 + · · · + cnyn

  Theorem: Suppose y1, y2, .., yn are solutions to y(n) + pn–1(x) y(n–1) + ··· + p1(x) y′ + p0(x) y = 0 on an interval J where the pi(x) are continuous.(a) If y1, y2, . . . , yn are linearly dependent then W(y1, y2, . . . , yn) = 0 at each point on the interval J.(b) If y1, y2, . . . , yn are linearly independent then W(y1, y2, . . . , yn) ≠ 0 at each point on the interval J.  Theorem: If y1, y2, . . . , yn are solutions to y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = 0, then so is y = c1y1 + c2y2 + · · · + cnyn for any choice of constants c1, c2, . . . , cn. Theorem: If y1, y2, . . . , yn are linearly independent solutions to y(n–1) + pn–2(x) y(n–2) + · · · · · + p1(x) y′ + p0(x) y = 0, then general solution is y = c1y1 + c2y2 + · · · + cnyn.

Ex. 1 Suppose that the following functions are all particular solutions to y(3) + 3y″ + 4y′ + 12y = 0. We would like to have a set of three linearly independent solutions. If this can be done then which three can we use? If it cannot be done then show that it cannot.y1 = sin(2x)y2 = cos(2x)y3 = 5 sin(2x) + 3 cos(2x)y4 = sin(x) cos(x)y5 = cos2(x) – sin2(x)y6 = e–3x

Ex. 2 Use the definition of linear dependence to show that the following functions are linearly dependent.f (x) = x2 + 3x, g(x) = 7x2, h(x) = 31x2 + 6x.

The Wronskian of the n functions f1(x), f2(x), . . . , fn(x):

1 2 3

1 2 3

1 2 3 1 2 3

( ) ( ) ( ) ( )1 2 3

, , , ,

n

n

n n

n n n nn

f f f f

f f f fW f f f f f f f f

f f f f

Theorem:Suppose y1, y2, ..., yn are solutions of y(n) + pn–1(x) y(n–1) + ··· + p1(x) y′ + p0(x) y = 0 on an interval J in which pn–1(x), pn–2(x), · · · p1(x), p0(x) are continuous.(a) If y1, y2, . . . , yn are linearly dependent then W(y1, y2, . . . , yn) = 0 for all x on the interval J.(b) If y1, y2, . . . , yn are linearly independent then W(y1, y2, . . . , yn) ≠ 0 for all x on the interval J.

Ex. 3 (a) y1 = sin(2x), y2 = cos(2x), y3 = e–3x are solutions to y(3) + 3y″ + 4y′ + 12y = 0. Use the Wronskian to determine whether or not they are linearly independent.

Ex. 3 (b) y1 = sin(2x), y2 = cos(2x), y3 = e–3x are solutions to y(3) + 3y″ + 4y′ + 12y = 0. Use the Wronskian to determine whether or not they are linearly independent.

Ex. 3 (c) y1 = x, y2 = x ln(x), y3 = x2 are solutions to x3y(3) – x2y″ + 2xy′ – 2y = 0. Use the Wronskian to determine whether or not they are linearly independent.

Suppose we can find three linearly independent solutions to y(3) + 3 y″ + 4y′ + 12y = 0.

Are we guaranteed a solution to the initial valued problem with given initial conditions of y(0) = 1, y′(0) = 2, y″(0) = 3? Why??

Theorem:Suppose pn–1(x), pn–2(x), · · · p1(x), p0(x) are continuous on an interval J which contains xo. If we can find n linearly independent solutions to the diff. eq.

y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = 0 then we are guaranteed a solution to the initial value problem of

y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = 0 y(x0) = b0 y′(x0) = b1 y″(x0) = b2 : :y(n)(x0) = bn

Theorem: If y1, y2, . . . , yn are linearly independent solutions to

y(n) + pn–1(x) y(n–1) + · · · + p1(x) y′ + p0(x) y = 0, then general solution is y = c1y1 + c2y2 + · · · + cnyn.

Nonhomogeneous linear Differential Equations: y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = f (x)

We solve a nonhomogeneous differential equation by first examining the corresponding homogeneous differential equation.

Review:

Theorem: Suppose that yc = c1y1 + c2y2 + · · · + cnyn is the general solution to the homogeneous differential equation y(n) + pn–1(x) y(n–1) + ··· + p1(x) y′ + p0(x) y = 0. Further suppose that yp is any particular solution to the nonhomogeneous differential equation y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = f (x). Then the general solution to the nonhomogeneous differential equation y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = f (x) is:

y = yp + yc = yp + c1y1 + c2y2 + · · · + cnyn

Ex. 4 (a) Verify that y = (–1/4) x is a particular solution to the diff. eq. y″ – 4y = x.

Ex. 4 (b) Verify that y = (–1/4) x + e2x is another particular solution to the diff. eq. y″ – 4y = x.

Ex. 4 (c) What is the general solution to the differential equation y″ – 4y = x ?

Ex. 4 (d) Find a solution to the initial value problem y″ – 4y = x, y(0) = 11, y′(0) = –2.

Ex. 5 (a) y = 2x is one solution to the differential equation y″ – 10y′ + 21y = 42x – 20. Use this to find the general solution.

Ex. 5 (b) y = 2x + 5e3x is another solution to the differential equation y″ – 10y′ + 21y = 42x – 20. Use this to find the general solution.

Section 2.3 Homogeneous Equations with Constant Coefficients

Find the general solution to y(3) + y″ – 17y′ + 15y = 0 

Recall:If r1 and r2 are distinct roots of the characteristic equation ar2 + br + c = 0, then and are linearly independent solutions to the differential equation ay″ + by′ + cy = 0, and thus is the general solution to ay″ + by′ + cy = 0.

11

r xy e 22

r xy e1 2

1 2r x r xy c e c e

Recall:If r1 and r2 are distinct roots of the characteristic equation ar2 + br + c = 0, then and are linearly independent solutions to the differential equation ay″ + by′ + cy = 0, and thus is the general solution to ay″ + by′ + cy = 0.

Now:

Theorem: If r1, r2, . . . , rk are distinct roots of the characteristic equation anrn + an–1rn–1 + · · · + a1r + a0 = 0, then are linearly independent solutions to the differential equation any(n) + an–1 y(n–1) + · · · + a1y′ + a0y = 0.

11

r xy e 22

r xy e1 2

1 2r x r xy c e c e

31 21 2 3, , , , kr x r xr x r x

ky e y e y e y e

Ex. 1 Find the general solution to y(3) + y″ – 17y′ + 15y = 0.

Ex. 2 Find the particular solution to y(3) + y″ – 17y′ + 15y = 0y(0) = 10y′(0) = 8y″(0) = 50

Ex. 3 Find the general solution to y(5) – 3y(4) + 2y(3) + 2y″ – 3y′ + y = 0.

Theorem: If r1 is a root of multiplicity k in the characteristic equation anrn + an–1rn–1 + · · · + a1r + a0 = 0, then are linearly independent solutions to the differential equation any(n) + an–1 y(n–1) + · · · + a1y′ + a0y = 0.

1 1 1 12 11 2 3, , , ,r x r x r x r xk

ky e y xe y x e y x e

Ex. 4 Find the general solution to y(9) – 25y(8) + 264y(7) – 1520y(6) + 5120y(5) – 9984y(4) + 10240y(3) – 4096y″ = 0.

Section 2.5 Nonhomogeneous Equations &

Undetermined Coefficients

Recall:

Theorem: Suppose that yc = c1y1 + c2y2 + · · · + cnyn is the general solution to the homogeneous differential equation y(n) + pn–1(x) y(n–1) + ··· + p1(x) y′ + p0(x) y = 0. Further suppose that yp is any particular solution to the nonhomogeneous differential equation y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = f (x). Then the general solution to the nonhomogeneous differential equation y(n) + pn–1(x) y(n–1) + · · · · · + p1(x) y′ + p0(x) y = f (x) is:

y = yp + yc = yp + c1y1 + c2y2 + · · · + cnyn

Recall:Ex. 5 (a) y = 2x is one solution to the differential equation y″ – 10y′ + 21y = 42x – 20. Use this to find the general solution.

How do we determine yp though?

The Method of Undetermined Coefficients

(Using an educated guess to determine yp)

Ex. 1 y″ + 2y′ – 3y = 3e2x

(a) Make a guess on what yp should look like for this differential equation? (What "form" should yp have?)

Ex. 1 y″ + 2y′ – 3y = 3e2x

(b) Determine, specifically, what yp is.

Ex. 1 y″ + 2y′ – 3y = 3e2x

(c) Give the general solution to this nonhomogeneous differential equation.

Ex. 2 y″ + 2y′ – 3y = 30 cos(x)(a) Make a guess on what yp should look like for this differential equation? (What "form" should yp have?)

Ex. 2 y″ + 2y′ – 3y = 30 cos(x)(b) Determine, specifically, what yp is.

Ex. 2 y″ + 2y′ – 3y = 30 cos(x)(c) Give the general solution to this nonhomogeneous differential equation.

Ex. 3 Find the general solution to 2y″ + 4y′ + 7y = x2

Ex. 4 Find the general solution to y″ – 4y = 2e2x

Ex. 4 Find the general solution to y″ – 4y = 2e2x

Ex. 5 Find the form of yp for the nonhomogeneous differential equation: y″ – 4y = 2xe2x

Ex. 6 Find the form of yp for the nonhomogeneous differential equation: y(3) + 9y′ = x sin(7x) + x2e2x

Ex. 7 Find the form of yp for the nonhomogeneous differential equation: (D – 2)(D – 5)3 y = xe2x + x2e5x + cos(x)

Lemma 1: For any function u(x) we have: (D – r)[u(x)erx] = (D [u(x)]) (erx)

Proof:

(D – r)[u(x)erx] = D[u(x)erx] – r[u(x)erx]

= D[u(x)] erx + u(x)D[erx] – ru(x)erx

= D[u(x)] erx + ru(x)erx – ru(x)erx

= D[u(x)] erx

Lemma 2: For any function u(x) we have: (D – r)k [u(x)erx] = (Dk [u(x)]) (erx)

Question: Find the form of yc for the differential equation (D – r)3 y = (–3 + 2x)erx

Solution: We compute yc : yc = c1erx + c2xerx + c3x2erx  

Our initial guess for yp : yp = Aerx + Bxerx  

Our adjusted guess for yp : yp = Ax3erx + Bx4erx

Why did this second guess work?This is the correct form for yp, but why? We have seen why our first guess at yp won't work. But why does throwing in higher powers of x into these terms give us the correct guess for yp?

Before we answer this we must first note the following:

Note that if we determined the values of A and B we would have our general solution to this nonhomogeneous equation be

y = c1erx + c2xerx + c3x2erx + Ax3erx + Bx4erx

Now let's look at the question again, this time without relying on our theorem from chapter 2 which indicates that throwing in higher powers of x into these terms give us the correct guess for yp .

Question: Find the form of yp for the differential equation (D – r)3 y = (–3 + 2x)erx

 Solution:

We transform this difficult nonhomogeneous diff. eq. into an easy homogeneous differential equation: 

Question: Find the form of yp for the differential equation (D – r)3 y = (–3 + 2x)erx

 Solution: 

If y is a solution to (D – r)3 y = (–3 + 2x)erx

then y is a solution to (D – r)2 [(D – r)3 y] = (D – r)2 [(–3 + 2x)erx] then y is a solution to (D – r)5 y = (D – r)2 [(–3 + 2x)erx] then y is a solution to (D – r)5 y = D2 [(–3 + 2x)] erx

then y is a solution to (D – r)5 y = 0 erx

then y is a solution to (D – r)5 y = 0

Thus y takes the following form: y = c1erx + c2xerx + c3x2erx + c4x3erx + c5x4erx

y = c1erx + c2xerx + c3x2erx + Ax3erx + Bx4erx

yc yp

Since we know that this first part comes from yc, the second part must come from yp. Therefore the form of yp is yp = Ax3erx + Bx4erx .

Ex. 8 Find the general solution to (D – 2)5 y = e2x + 3xe2x

The Method of Variation of Parameters

Try to use the method of undetermined coefficients to solve: y″ + y = tan(x)

Suppose we are given the differential equation y″ + P(x) y′ + Q(x) y = f (x) and we have determined that yc = c1y1 + c2y2. It turns out that yp can be formed from y1 and y2. In fact there exists some functions u1(x) and u2(x) so that yp = u1y1 + u2y2. 

We are given the differential equation y″ + P(x) y′ + Q(x) y = f (x), and we have determined that yc = c1y1 + c2y2. Suppose yp = u1y1 + u2y2 is a particular solution to the nonhomogeneous equation (where u1 and u2 are some functions unknown to us). We must determine what u1 and u2 are.

yp = u1y1 + u2y2

yp′ = (u1′ y1 + u1y1′ ) + (u2′ y2 + u2y2′ ) = (u1y1′ + u2y2′ ) + (u1′ y1 + u2′ y2 ) 

We shall now impose a restriction of u1′ y1 + u2′ y2 = 0  

So, yp′ = u1 y1′ + u2 y2′  

yp″ = (u1 y1′ + u2 y2′ )′ 

yp″ = (u1′ y1′ + u1 y1″ ) + (u2′ y2′ + u2 y2″ ) 

yp″ = u1′ y1′ + u2′ y2′ + u1 y1″ + u2 y2″ 

yp″ = u1′ y1′ + u2′ y2′ + u1[–P(x) y1′ – Q(x) y1] + u2 [–P(x) y2′ – Q(x) y2] 

yp″ = u1′ y1′ + u2′ y2′ – P(x) [u1 y1′ + u2 y2′] – Q(x) [u1y1 + u2y2] 

yp″ = u1′ y1′ + u2′ y2′ – P(x) [yp′ ] – Q(x) [yp] 

yp″ + P(x) yp′ + Q(x) yp = u1′ y1′ + u2′ y2′  

f (x) = u1′ y1′ + u2′ y2′

Now we have two conditions placed on u1 and u2: u1′ y1 + u2′ y2 = 0u1′ y1′ + u2′ y2′ = f (x)

Now we have two conditions placed on u1 and u2: u1′ y1 + u2′ y2 = 0u1′ y1′ + u2′ y2′ = f (x)

Cramer's rule states that we can solve this for u1′ and u2′ as long as 1 2

1 2

0.y y

y y

Now we have two conditions placed on u1 and u2: u1′ y1 + u2′ y2 = 0u1′ y1′ + u2′ y2′ = f (x)

Cramer's rule states that we can solve this for u1′ and u2′ as long as

Cramer's rule states

1 2

1 2

0.y y

y y

2

21

1 2

1 2

0

( )

y

f x yu

y y

y y

1

12

1 2

1 2

0

( )

y

y f xu

y y

y y

Now we have two conditions placed on u1 and u2: u1′ y1 + u2′ y2 = 0u1′ y1′ + u2′ y2′ = f (x)

Cramer's rule states that we can solve this for u1′ and u2′ as long as

Cramer's rule states

1 2

1 2

0.y y

y y

2

21

1 2

1 2

0

( )

y

f x yu

y y

y y

2

1 2

( )( , )f x y

W y y

1

12

1 2

1 2

0

( )

y

y f xu

y y

y y

1

1 2

( )( , )f x y

W y y

Now we have two conditions placed on u1 and u2: u1′ y1 + u2′ y2 = 0u1′ y1′ + u2′ y2′ = f (x)

Cramer's rule states that we can solve this for u1′ and u2′ as long as

Cramer's rule states

Now, integrate to determine u1 and u2.

1 2

1 2

0.y y

y y

2

21

1 2

1 2

0

( )

y

f x yu

y y

y y

2

1 2

( )( , )f x y

W y y

1

12

1 2

1 2

0

( )

y

y f xu

y y

y y

1

1 2

( )( , )f x y

W y y

21

1 2

( )( , )f x yu dx

W y y

12

1 2

( )( , )f x yu dx

W y y

Ex. 9 Use variation of parameters to solve: y″ + y = tan(x)

Ex. 9 Use variation of parameters to solve: y″ + y = tan(x)

Ex. 10 Use variation of parameters to solve: y″ – 4y = 2e2x

Ex. 10 Use variation of parameters to solve: y″ – 4y = 2e2x

Section 2.4 Mechanical Vibrations

Forces Involved: FS = –kx (k > 0) Restorative Force (Hooke's Law)FR = –cx′ (c > 0) Force due to dashpot

Total Force = FR + FS

mx″ = –cx′ – kx

mx″ + cx′ + kx = 0

Later we will include external force and get the differential equation mx″ + cx′ + kx = F(t)

Terminology: If c = 0 the motion is undamped If c > 0 the motion is damped If F(t) = 0 the motion is free If F(t) ≠ 0 the motion is forced

Free Undamped Motionmx″ + kx = 0

::

x(t) = A cos(ωot) + B sin(ωot) where A and B are some constants and : :

x(t) = C cos(ωot – α) where C and α are some constants and

ko m

ko m

Free Undamped Motionmx″ + kx = 0

::

x(t) = A cos(ωot) + B sin(ωot) where A and B are some constants and : :

x(t) = C cos(ωot – α) where C and α are some constants and

ko m

ko m

Free Undamped Motionmx″ + kx = 0

::

x(t) = A cos(ωot) + B sin(ωot) where A and B are some constants and : :

x(t) = C cos(ωot – α) where C and α are some constants and

For this last equation we have the following interpretations:C Amplitude

ωo (Circular) Frequency

α Phase Angle

Frequency

Period

ko m

ko m

2o

2o

Ex. 1 A body with mass m = (1/2)kg is attached to the end of a spring that is stretched 2m by a force of 100 N. It is set in motion with initial position xo = 0.5m and initial velocity vo = –10m/s. Find the position function of the body as well as the amplitude, frequency, period of oscillation, and phase angle of its motion.

Ex. 1 A body with mass m = (1/2)kg is attached to the end of a spring that is stretched 2m by a force of 100 N. It is set in motion with initial position xo = 0.5m and initial velocity vo = –10m/s. Find the position function of the body as well as the amplitude, frequency, period of oscillation, and phase angle of its motion.

Damped Free Motionmx″ + cx′ + kx = 0 

Damped Free Motionmx″ + cx′ + kx = 0 The characteristic equation has roots:

2 42

c c kmm

Damped Free Motionmx″ + cx′ + kx = 0 The characteristic equation has roots:

Whether these roots are real, complex, or repeated depends on the sign of c2 – 4km.So, the sign of c2 – 4km really amounts to whether

c = , c < , or c >

2 42

c c kmm

4km 4km 4km

Damped Free Motionmx″ + cx′ + kx = 0 The characteristic equation has roots:

Whether these roots are real, complex, or repeated depends on the sign of c2 – 4km.So, the sign of c2 – 4km really amounts to whether

c = , c < , or c >

Critical damping: (results in repeated roots)

2 42

c c kmm

4km 4km 4km

4c km1 1

1 2r x r xy c e c xe

2 4 6 8 10

-0.1

-0.05

0.05

0.1

0.15

Damped Free Motionmx″ + cx′ + kx = 0 The characteristic equation has roots:

Whether these roots are real, complex, or repeated depends on the sign of c2 – 4km.So, the sign of c2 – 4km really amounts to whether

c = , c < , or c >

Overdamping: (results in distinct real roots)

2 42

c c kmm

4km 4km 4km

4c km1 2

1 2r x r xy c e c e

2 4 6 8 10

-0.1

-0.05

0.05

0.1

0.15

0.2

Damped Free Motionmx″ + cx′ + kx = 0 The characteristic equation has roots:

Whether these roots are real, complex, or repeated depends on the sign of c2 – 4km.So, the sign of c2 – 4km really amounts to whether

c = , c < , or c >

Underdamping: (results in complex roots)

y = eAx(c1 cos(Bx) + c2 sin(Bx))

2 42

c c kmm

4km 4km 4km

4c km

2 4 6 8 10

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

Section 2.6 Forced Oscillations & Resonance

mx″ + cx′ + kx = F(t)

where F(t) = Fo cos(ωt) or F(t) = Fo sin(ωt)

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)  

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)  xc = c1 cos(ωot) + c2 sin(ωot) (recall )  If ω ≠ ωo then we shall guess that xp = A cos(ωt).

ko m

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)  xc = c1 cos(ωot) + c2 sin(ωot) (recall )  If ω ≠ ωo then we shall guess that xp = A cos(ωt).Using our guess of xp = A cos(ωt) in the diff. eq. mx″ + kx = Fo cos(ωt)

we find that

ko m

2 2o

o

FAm

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)  xc = c1 cos(ωot) + c2 sin(ωot) (recall )  If ω ≠ ωo then we shall guess that xp = A cos(ωt).Using our guess of xp = A cos(ωt) in the diff. eq. mx″ + kx = Fo cos(ωt)

we find that

So,

ko m

2 2o

o

FAm

2 2coso

po

Fx tm

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)  xc = c1 cos(ωot) + c2 sin(ωot) (recall )  If ω ≠ ωo then we shall guess that xp = A cos(ωt).Using our guess of xp = A cos(ωt) in the diff. eq. mx″ + kx = Fo cos(ωt)

we find that

So,

Thus, the general solution is

ko m

2 2o

o

FAm

2 2coso

po

Fx tm

2 2cos coso

oo

Fx C t tm

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)  xc = c1 cos(ωot) + c2 sin(ωot) (recall )  If ω ≠ ωo then we shall guess that xp = A cos(ωt).

Question - What happens when ω = ωo?Answer - Resonance

ko m

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)

xc = c1 cos(ωot) + c2 sin(ωot) (recall ) Suppose ω = ωo .

Our guess for xp would be: xp = At cos(ωot) + Bt sin(ωot).

ko m

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)

xc = c1 cos(ωot) + c2 sin(ωot) (recall ) Suppose ω = ωo .

Our guess for xp would be: xp = At cos(ωot) + Bt sin(ωot). : A = 0

ko m

0

02FBm

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)

xc = c1 cos(ωot) + c2 sin(ωot) (recall ) Suppose ω = ωo .

Our guess for xp would be: xp = At cos(ωot) + Bt sin(ωot). : A = 0

So,

ko m

0

02FBm

0

0

sin2p o

Fx t tm

Undamped Forced Oscillations

mx″ + kx = F(t) Let's examine mx″ + kx = Fo cos(ωt)

xc = c1 cos(ωot) + c2 sin(ωot) (recall ) Suppose ω = ωo .

Our guess for xp would be: xp = At cos(ωot) + Bt sin(ωot). : A = 0

So,

Thus, the general solution is

ko m

0

02FBm

0

0

sin2p o

Fx t tm

0

0

cos sin2o o

Fx C t t tm