Solve the PDE $2U_x -3U_y=x$

Question

Solve the PDE

$2U_x -3U_y=x$

$u=u(x,y)$

I wanted to make sure my solution is correct:

characteristic lines: $3x+2y=d$

Change of variables

$w=3x+2y$

$z=y$

The PDE then becomes the ODE: $V_z= w+2z$

Thus the solution is $u(x,y)= (3x+2y)y + y^2 + C(3x+2y)$

As a start, the solution you give does not satisfy the differential equation (you will find $2U_x-3U_y = -12y - 9x\ne x$). — Doubt, Feb 17 '13 at 02:54

Mhenni Benghorbal · Answer 1 · 2013-02-17T03:53:30.043

1

Just use the method of characteristic. Here is the final answer

$$ u \left( x,y \right) =\frac{1}{4}\,{x}^{2}+{F} \left( y+\frac{3}{2}\,x \right). $$

edited Feb 17 '13 at 03:53

answered Feb 17 '13 at 03:29

Mhenni Benghorbal

score 0 · Answer 2 · answered Feb 17 '13 at 03:57

0

Simply use method of characteristics.

answered Feb 17 '13 at 03:57

Raman Gupta

2

Welcome to MSE! I would recommend adding more details to your answer as it may not be clear to the OP. Regards – Amzoti Feb 17 '13 at 04:32

2 Answers2