# The solution of the PDE

1. Jul 10, 2010

### fderingoz

find the general solution of yux+xuy=yu+xex ( the solution is in the form of u(x,y)=yex+f(y2-x2)ex )
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.

Last edited: Jul 10, 2010
2. Jul 10, 2010

### csopi

3. Jul 10, 2010

### fderingoz

i tried the method of characteristics,but i can not find the solution

4. Jul 10, 2010

### 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

5. Jul 12, 2010

### 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.