Solving Partial Differential Equation

[Peter Alexander]

1. The problem statement, all variables, and given/known data
Task requires you to solve a partial differential equation $$u_{xy}=2yu_x$$ for $u(x,y)$. A hint is given that a partial differential equation can be solved in terms of ordinary differential equations.
According to the solution sheet, the result should be $u(x,y)=e^{y^2}f(x) + g(y)$ where $f,\:g$ seem to be two arbitrary functions.

Homework Equations

I don't think there are any relevant equations to explicitly mention.

The Attempt at a Solution

I believe that a solution to the task hides somewhere in the example I've solved before. Given a partial differential equation $$u_{xx}=0$$ which has to be solved for $u(x,y)$, I've simply assumed that $$u(x,y) \rightarrow u(x)$$which evidently produced a second order linear homogeneous ODE. Then, I would simply use $u(x)=e^{ax}$ and its second order derivative to find a general solution $u(x,y) = c_1(y)+xc_2(y)$.

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If I would want to produce the same ODE in my example, then I guess I should be looking at the partial derivatives and what they have in common. I can see that both sides of the equation include a partial derivative with respect to $x$, so can I make a substitution $u_x = v$ or something like that?

But a problem with this approach is t

I am stuck solving this task. I would like to ask for some help, but I do not expect anyone to solve my homework instead of me. Instead, all I'm asking for is some guidance or tips I can use to solve the task myself.

PS: I hope you're having a wonderful Monday!

 

[phyzguy]

Try calling the function $u_x = f$. Then the equation reduces to the ODE $f_y = 2 y f$. Solve this for f(y), then plug this back into $u_x = f$ to get the x dependence.

 

[Peter Alexander]

Thank you so much! I actually managed to find a solution this way, thanks!