Solving Partial Differential Equation

  • #1
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!
 

Answers and Replies

  • #2
phyzguy
Science Advisor
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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.
 
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Likes Peter Alexander and Chestermiller
  • #3
Thank you so much! I actually managed to find a solution this way, thanks!
 

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