General Solution of PDE yux+xuy=yu+xex: Existence and Infinite Solutions

[fderingoz]

find the general solution of yu_x+xu_y=yu+xe^x ( the solution is in the form of u(x,y)=ye^x+f(y²-x²)e^x )
if at first the value of u(x,y) on the upper half of hyperbola (that is y>=1) has been given as φ,show that if φ has not been given as a special form there is no solution.find that special form of φ and show there is infinite solution in this situation.

 

[csopi]

Try the method of characteristics. It should work.
In case you're not familiar with it:
http://en.wikipedia.org/wiki/Method_of_characteristics

 

[fderingoz]

thanks for your advice
i tried the method of characteristics,but i can not find the solution

 

[hunt_mat]

The characteristic equations are
$$\dot{x}=y,\quad\dot{y}=x,\quad\dot{u}=yu+xe^{x}$$
then the characteristic are given as $$dy/dx=x/y$$. Then this integrates up to $$f(x,y)=C$$. Then use $$du/dx=\dot{u}/\dot{x}$$ and integrate up.

Mat

 

[fderingoz]

i found the general solution of the equation. thanks for your helps
but i can't understand anything rest of the question.i am waiting for your helps.