Toronto Math Forum
APM3462019 => APM346Lectures & Home Assignments => Home Assignment 2 => Topic started by: Zhiman Tang on January 18, 2019, 11:53:36 PM

the expression on the right hand contains both x and y,
the characteristic line is 3xy=c
then, we evaluate du = xydx. in this step, we have to replace y by 3xc before integrating both sides.
My question is, why we have to do so?
And after integrte both sides, why we have to replace c back by 3xy?

Please learn how to post math properly (http://forum.math.toronto.edu/index.php?topic=610.0) Also, asking for help, copy the problem.

We can make the substitution $y = 3x  C$ because we are restricting $u(x, y)$ to the characteristic curves, so I believe we can treat $y$ as equal to $3x  C$ when finding the general solution. We do this because we need the $xydx$ totally in terms of $x$ or we will not be able to integrate both sides. After integrating, we have to get rid of $C$ by replacing it with $3xy$ again because we want our final solution $u$ to be a function of $x$ and $y$, not of $C$. $C$ is just a constant but it is still in terms of $x, y$ by the characteristic curves.
$C$ is a constant only along integral curves. V.I.

Would we be incorrect to define $C = y3x$ since saying $y = 3xC$ is essentially equivalent to $y = 3x+C$ for arbitrary $C$? This is what I did in my solution.