690 CHAPTER 34. SOME PARTIAL DIFFERENTIAL EQUATIONS

3. Find the solution to the initial boundary value problem

utt = 4uxx,u(0, t) = 0,u(5, t) = 0,u(x,0) = 3x(x−5) ,ut (x,0) = x+1

4. Find the solution to the initial boundary value problem

utt = 4uxx,u(0, t) = 0,u(2, t) = 0,u(x,0) = −x(x−2) ,ut (x,0) = x2

5. Describe how to solve the initial boundary value problem

utt +2ut = 2uxx,u(0, t) = 0,u(5, t) = 0,u(x,0) = −x(x−5) ,ut (x,0) = x2

Hint: You might consider defining w = e2tu and see what equation is solved by w.

6. Find the solution to the initial boundary value problem

ut −2u = uxx,u(0, t) = 0,u(5, t) = 0,u(x,0) = x

Hint: It is like before. You get eigenfunctions and match coefficients.

7. Find the solution to the initial boundary value problem

ut = uxx +(cosx) ,u(0, t) = 0,u(2, t) = 0,u(x,0) = 1− x

8. Find the solution to the initial boundary value problem

ut = 2uxx +(x−1) ,u(0, t) = 0,u(4, t) = 0,u(x,0) = 1

9. Find the solution to the initial boundary value problem

ut = 5uxx +(x−1) ,u(0, t) = 0,u(1, t) = 0,u(x,0) = x+1

10. Find the solution to the initial boundary value problem

ut = 2uxx,u(0, t) = 0,u(5, t) = 0,

u(x,0) =x for x ∈

[0, 5

4

]0 if x ∈ ( 5

4 ,5]

11. Find the solution to the initial boundary value problem

ut = 3uxx,u(0, t) = 0,u(3, t) = 0,

u(x,0) =1 for x ∈ [0,1]0 if x ∈ (1,3]