21b Review Differential equations
1. (a) Find the general solution to the di↵erential equation f00(t)+3f0(t)4f(t) = 0.
(b) Find the general solution to the di↵erential equation f00(t)+3f0(t)4f(t)=2et.
2. Solve the following PDEs (assuming fis defined on [0,⇡] and satisfies the boundary condition f(t, 0) = f(t, ⇡) = 0)
(a) ft=fxx +fwith the initial condition f(0,x)=4sin(3x)8sin(11x)
(b) ftt =fxx +fwith the initial condition f(0,x)=6sin(x) + sin(13x) and ft(0,x)=0
3. The wave equation @2f
@t2=c2@2f
@x2we have been looking at models the movement of something like a
guitar string, but it does not take into account any damping forces such as air resistance.
The partial di↵erential equation @2f
@t2+@f
@t=@2f
@x2is an example of a damped wave equation, which
models a guitar string which is subject to damping forces.
(a) Find the general solution of the damped wave equation @2f
@t2+@f
@t=@2f
@x2satisfying the boundary
conditions f(t, 0) = f(t, ⇡) = 0. (Please use the method we used to derive the solution of the wave
equation; start by writing f(t, x)=
1
X
k=1
bk(t)sin(kx) and finding di↵erential equations satisfied by
the coefficients bk(t).)
(b) What can you say about lim
t!1 f(t, x)? Does this agree with your intuition, given what is being
modeled?
4. (a) Find the Fourier series of f(x)=x2. (Note: Zx2cos kxdx =x2sin kx
k+2xcos kx
k22sinkx
k3+C.
This integral can be evaluted by using integration by parts twice.)
(b) Use Parseval’s Theorem to evaluate the sum
1
X
k=1
1
k4.
5. Which of the following is a linear space?
(a) all ~xwith A~x=~
0
(b) all ~xwith A~x=~
b
(c) En, the polynomials of degree nwith only even powers of t
(d) the set of n⇥nmatrices with determinant zero
(e) the set of n⇥nmatrices with trace zero
(f) all f(t)withf00(t)+af0(t)+bf (t)=0
(g) all f(t)withf00(t)+af 0(t)+bf(t)=g(t)
1
