I am trying to relearn a bajillion concepts and learn the symmetry method of solving differential equations.
I am working through an example so that I can check my answers.
However, the step by step method is not there, so I am stuck.
I have found system of determining equations and now need to solve each one.
Please can someone show me or direct me to a textbook that will explain how to solve the following:
$ \xi_{yy} + \frac{1}{y} \xi_{y} = 0 $
and
$ \eta_{yy} - 2\xi_{xy} - \frac{1}{y} \eta_{y} + \frac{1}{y^2} \eta = 0 $
for $\eta(x,y)$ and $\xi(x,y)$
Many thanks,
Sincerely,
Sarah
Start with the equation for $\xi$. We have,
$$\xi_{yy} +\frac{1}{y}\xi_y = 0\\ \implies y\xi_{yy} + \xi_y=0$$
which is a Cauchy-Euler equation with solution $\xi(x,y) = c_1(x)\log(y) + c_2(x)$.
From here we calculate $\xi_{xy} = \frac{c_1'(x)}{y}$ and substitute into the equation for $\eta$,
$$\eta_{yy} - 2\frac{c_1'(x)}{y} -\frac{1}{y}\eta_y + \frac{1}{y^2}\eta=0$$
This is again an ode in the variable $y$ (in fact, another Cauchy-Euler equation).