A solution of the relatively complex problem led to the following non-linear PDE of the first order
$$\left(\frac{\partial t}{\partial x}\right)^{2}+\left(\frac{\partial t}{\partial y}\right)^{2}=\frac{t^{2}(t-1)^{2}(t^{2}+2t+5)^{2}}{(y(t^{3}+3t-10)+t(t^{2}+5))^{2}}.$$ Unfortunately, the right side is relatively complex and can not be expressed in the form of $f(y)g(t)$ that is suitable for the substitution $$dz=\frac{dt}{\sqrt{g(t)}},\qquad z=\int\frac{1}{\sqrt{g(t)}}dt,$$
transfoming PDE $$\left(\frac{\partial t}{\partial x}\right)^{2}+\left(\frac{\partial t}{\partial y}\right)^{2}=f(y)g(t),$$
into a more simple form of $$\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=f(y).$$
Is there any reasonable substitution for PDE of the form $$\left(\frac{\partial t}{\partial x}\right)^{2}+\left(\frac{\partial t}{\partial y}\right)^{2}=f(y,t), $$ leading to relatively “simple” Lagrange-Charpit equations that can be applied to the above-mentioned PDE?
Thanks for your help.