# Thread: Find the general solution to the equation 2u_x + u_xy = 0

1. ## Find the general solution to the equation 2u_x + u_xy = 0

where u_x is u(x,y) differentiated with respect to x and u_xy is u(x,y) twice differentiated with respect to x and with respect to y

How would I go about solving this?

Thanks for any help.

2. ## Re: Find the general solution to the equation 2u_x + u_xy = 0

I'm pretty sure I've done it now actually.

u = e^(ax+2y)

3. ## Re: Find the general solution to the equation 2u_x + u_xy = 0

What you present is just one solution. Here are two more

(1) $\displaystyle u = e^{-2y} \sin x$

(2) $\displaystyle u = x e^{-2y} + y^2$

Why not let $\displaystyle v = u_x$ and see where that gets you.

4. ## Re: Find the general solution to the equation 2u_x + u_xy = 0

Originally Posted by Danny

Why not let $\displaystyle v = u_x$ and see where that gets you.
This gives 2v + v_y = 0

I'm not really sure what to do

5. ## Re: Find the general solution to the equation 2u_x + u_xy = 0

Separate and integrate noting that you'll get a function of integration.

6. ## Re: Find the general solution to the equation 2u_x + u_xy = 0

Since only differentiation with respect to y is involved, you can solve that as the ODE
$\displaystyle \frac{dv}{dy}= -2y$. Of course, the "constant" may be a function of x. What does that give you for v? What does that make the orginal equation?

Remember that the general solution to a partial differential equation may involve unknown functions of the variables rather than unknown constants.

7. ## Re: Find the general solution to the equation 2u_x + u_xy = 0

Originally Posted by HallsofIvy
Since only differentiation with respect to y is involved, you can solve that as the ODE
$\displaystyle \frac{dv}{dy}= -2y$.
surely you mean dv/dy = -2v ?

8. ## Re: Find the general solution to the equation 2u_x + u_xy = 0

Originally Posted by feyomi
surely you mean dv/dy = -2v ?
I'm sure that's what HoI meant.