Toronto Math Forum
APM3462022S => APM346Lectures & Home Assignments => Chapter 2 => Topic started by: Yifei Hu on February 02, 2022, 04:35:19 PM

The problem asks for general solution of the equation $U_t+yU_xxU_y=0; U(0,x,y)=f(x,y)$
I proceed as usual:
$$\frac{dt}{1}=\frac{dx}{y}=\frac{dy}{x}=\frac{du}{0}$$
Integrate and this gives: $x^2+y^2=C$, $t\int \frac{1}{\sqrt{cx^2}}dx=D$
Hence, I conclude that $U=\phi(C,D)=\phi(x^2+y^2,t\int \frac{1}{\sqrt{cx^2}}dx)$. However, this involves an integral that I can not calculated by hand, can anyone give me a hint on how to do this integral?
Also, I see that in the solution we can also solve this system with a nice trigonometry form $U=f(xcos(t)ysin(t),xsin(t)+ycos(t))$but the solution does not specify how to reach that, can anyone shed lights on how the solution is reached?
Thanks in advance.

 If you do not know this integral you need to refresh Calcuus I. one of basic integrals. Or have a table of basic integrals handy.
 Since $x^2+y^2=c^2$ is a circle, you can substitute $x=c\cos(s)$ and $y=c\sin(s)$ and then observe that $s=Ds$. It gives you the answer, less nicely looking than the one you wrote.
 Expressing $x, y$ through $t,c,d$ you can express $C=c\cos(d)$ and $D=c\sin(d)$ through $x,y,t$ which would give you that nice answer.
Write \cos , \sin , \log .... to produce proper (upright) expressions with proper spacing