I need help in solving this problem
Find the general solution of the partial differential equation $$x(z+2a)p+(xz+2yz+2ay)q=z(z+a),$$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$. Also find its particular integral surface which passes through the given curve $y=0$, $z^2=4ax$.
I know this is a Lagrange's linear equation and that I require two general solution of this equation to find the Integral surface.
I got the first general solution like this $$\frac{dx}{x(z+2a)}=\frac{dy}{xz+2yz+2ay}=\frac{dz}{z(z+a)}$$ comparing 1st and 3rd we get $$\frac{dx}{x}=\frac{(z+2a)dz}{z(z+a)}$$ Integrating this we get the first solution $$\frac{z^2}{x(z+a)}=c_1.$$ I just can not find the second solution , I have tried substituting $$x=\frac{z^2}{c_1(z+a)}$$ while equating the 2nd and 3rd equation but it gets much more complex.
Can you find the second solution?