A Challenging PDE question

Dear users of MHF, I've found two PDE question very challenging, would you guys mind to give some hints?

First question:

1) (-x) Ux + (2y) Uy = x^2 + y^2

a) find the characteristic curve (I solved)

b) solve the above equation by conditions U(x,1)= x^2 , u(x,-1) = x.

second question:

2)Ux + Uy= x - y + U

condition : U(0,y)= e^(-y^2).

for the second question I tried U = x^2 - (xy) + f(x-y) e^(x)

but I yielded a result like x - y + f(x-y)e^(x)....

the questions just sound so impossible for me...

and the condition in 1b was so tricky...

Re: A Challenging PDE question

Did you obtain tbe general solutions

(1) ?

In (2) you mentioned you tried Where did this come from?

Re: A Challenging PDE question

Re: A Challenging PDE question

For the first part, subs into your solution, set this equation the the IC and solve for . For example,

so .

Let so . This then defines . Similarly for the second IC.

For the second question, do you know the method of characteristics?

Re: A Challenging PDE question

With method of characteristic, I know how to get for Ux + Uy=0 u= f(x-y)

(at work, sorry for short reply)

Re: A Challenging PDE question

OK. Your answer above kinda makes sense. You can do one of two things. If you let

then your PDE

becomes . Then do what you did.

OR you can solve the charactistic equations

.