PDE: Method of characteristics question

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Homework Statement
x ut +ux =0
intial condition
u(x,0)=f(x)

1. Find the characteristics curves
2. What area of the xt-plane do u expect a solution
3. Find solution when f(x)=cos x
4.Now u(x,0)=f(x) (again), Find the level curves of u i.e for each c find the set Lc={(x,t):u(x,t)=f(c)}
5. Show there is not solution for u(x,0)=sin x
6. For what function is there a solution for u(x,0)=f(x), Then what the soution for u(x,y)?

The attempt at a solution
Can anyone check my solution to tell me if this is right and advise me on how to do this correctly. I think it wrong leading to incorrect answers everywhere else.
1. dt/dx = x
y=x2 +C

2. only on the x=0 since it can't apss throught the charactistics curve more than once

3. From initial
x(0,s) =s x(k,s) =s+k
t(0,s)=0 t(k,s) =xk
u(0,s)= cos s u(k,s) =cos s

u= cos (x/(x-y))
4. Not sure what 4 is asking can anyone head point me in the right direction?
5. I dont see how this is any different from the cos equation in part 3,
I end up getting u= sin(x/(x-y)) by the same method, but there not meant to be a solution
6. Any help is appreciated
 

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