Solve Olimpiad Problem: Draw u(5,x) Graph

[Mechmathian]

I am having problems on this "said to be easy" olimpiad problem.
If anyone can give me some hints, I'd be greatful!
(unfortunately I could not get the subscripts to work, that is why everything that is superscribed should be lowered)
u$$_{tt}$$ = a(x)*u$$_{xx}$$. x$$\in R$$

a(x) = 1 (if x>0)
a(x) = 2 (if x<=0)

u|$$_{t=0}$$ = b(x) u$$_{t}$$|$$_{t = 0}$$ = 0

b(x)= 0 when x$$\notin$$ [0, 1]
b(x) = 2x-2 when x$$\in [1, 1.5]$$
b(x) = -2x+4 when x$$\in[1.5, 2]$$

The problem is to draw the graph of the function u(t, x) when t = 5.

 

[Mechmathian]

I think that generalized functions have to be used here, although I am not 100% sure..

 

[Haelfix]

Whats U^tt. Is that supposed to be a double time derivative?

 

[Mechmathian]

yes, the indexes need to be a the bottom

 

[Mechmathian]

The thing is that I can solve the problem for any of the intervals ((-inf, 1), (1, 1.5), (1.5, inf)). But don't know how to get the general solution