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**Homework Statement**

x u

_{t}+u

_{x}=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 L

_{c}={(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=x

^{2}+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