Author Topic: Homework 1: Problem 3 (20)  (Read 636 times)

Jeannette Wong

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Homework 1: Problem 3 (20)
« on: January 30, 2018, 07:38:00 PM »
Find the general solution to the following equation:
Uxy = 2Ux

Can someone help me with this question please?

Jingxuan Zhang

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Re: Homework 1: Problem 3 (20)
« Reply #1 on: January 30, 2018, 08:04:03 PM »
$$u_{xy}=2u_{x} \implies u_{y}=2u+\varphi_{y}(y) \implies (e^{-2y}u)_{y}=e^{-2y}\varphi_{y}(y) \implies u=\varphi(y)+2e^{2y}(\int^{y}\varphi(s)e^{-2s}\,ds+\psi(x))$$
But it is really an ODE. If I have mistaken then please inform me. I have used intergration by part and standard ODE technique.

I made a lost of mistakes when I first post these, including: forget the arbitrary function after last integration wrt y; messed up the order of multiplication.

Edit:

Really such a mess is no better than something wrong. Sub in $v=u_{x}$ immediately $v=\varphi_{x}(x)e^{2y}\implies u=\psi(y)+\varphi(x)e^{2y}$.
« Last Edit: February 01, 2018, 08:27:21 AM by Jingxuan Zhang »

Victor Ivrii

It is correct, but overcomplicated). Simpler: denote $u_x=v$ and solve ODE