I am trying to learn how to solve an ODE using symmetry. I have got to the point where I have the generators, now I need to use them to get canonical coordinates, $r$ and $s$ where $r(x,y)$ and $s(x,y)$.
If $\eta(x,y)=xy$ and $\xi(x,y)=x^2$ Then the equations to solve in order to determine $r$ and $s$ are :
$x^2r_x + xyr_y =0$
and
$x^2s_x + xys_y=1$
however, I don't know how to move forward from here.
It has been eons since I have solved differential equations. Please can someone walk me through this?
UPDATE: I have tried to refresh my memory wrt to solving PDEs. Here is my attempt:
For the first equation I have the following:
divide by $x^2$ to get
$\frac{\partial r}{\partial x} + \frac{y}{x}\frac{\partial r}{\partial y}=0$
so I need to solve $\frac{dy}{dx}=\frac{y}{x}$
if I separate variables and integrate I get
$\ln y =\ln x+C$
$\ln(y/x)=C$
$\frac{y}{x}=C$
Does this mean which means $r(x,y) = \frac{y}{x}$ ?
I need to know if this is correct and I need help solving the second equation to find $s(x,y)$.
Thanks