Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Find the parametric solution of the PDE

$$xu_x -xyu_y - u=0$$

which follows the side condition $u(s^2, s)=s^3$

The solution uses another method rather than finding the general solution using parametrization of $x,y$ into $s,t$.

It says $X(s,t)=Ae^t$ and $Y(s,t)=Be^{-xt}$ and $U(s,t)=Ce^t$.

(I understand that they found $X(s,t)$ using $dx/dt$. I can't seem to solve $Y(s,t)$ and $U(s,t)$.

I also don't undestand how to find the constans A,B and C after.

share|cite|improve this question
What's $y_y$? I suppose it should be $u_y$. – Kaster Feb 18 '13 at 0:09
yes thank u for the correction @kaster – Yenny Chen Feb 18 '13 at 0:45
$u(x=s^2,y=s)=s^3$ or $u(y=s^2,x=s)=s^3$ ? – doraemonpaul Apr 21 '13 at 1:39




Follow the method in

$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$

$\dfrac{dy}{dt}=-y$ , letting $y(0)=y_0$ , we have $y=y_0e^{-t}=y_0e^{-x}$

$\dfrac{du}{dt}=\dfrac{u}{x}=\dfrac{u}{t}$ , we have $u(x,y)=tf(y_0)=xf(ye^x)$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.