Show that $u(x,y)=f(2y+x)+g(2y-x^2)$ is a general solution of the equation $$u_{xx}-1/xu_{x}-x^2u_{yy}=0$$
Finding $u_x, u_t, u_{yy}$ , we substitute them to the Pde and confirm equality, right?
or how to show it?
Asked
Active
Viewed 59 times
2
-
2If you substitute the expression into the differential equation, you will show that the expression is a solution of the DE. You still ahve to prove that all solutions of the DE have tis form. – TZakrevskiy Mar 24 '17 at 09:31
-
I understand thing you said, now. All right, How can we show it? Could you help me? – Mar 24 '17 at 10:04
-
Judging by the proposed form of the solution, the method characteristics would be a good start. Otherwise, by the same reasoning, you might try to make a change of variables $s = 2y+x$, $t=2y-x^2$, find where this change of variables has sense, and then rewrite the differential equation it terms of derivatives with respect to $s$ and $t$. – TZakrevskiy Mar 24 '17 at 10:07
-
how do we select $s$ and $t$? by looking function given $f$ ? or if it is not given? – Mar 24 '17 at 11:59
-
by looking at the arguments of $f$ and $g$. – TZakrevskiy Mar 24 '17 at 12:10