Did you obtain tbe general solutions
(1) ?
In (2) you mentioned you tried Where did this come from?
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...
Thanks danny, definitely I obtained (1), but when I try to use the condition given, I can't figure out how can I get u(x,-1)=x .
for 2) I only know that or usually results in , how ever the x-y parts I find it so hard...
if i let ,
,
but then applying the condtion given I can't get the answer
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?