y''(x)=-\frac{y'(x) \left(2 f(x) g(x) g'(x)^2-\left(g(x)^2-1\right) \left(2 f'(x) g'(x)+f(x) g''(x)\right)\right)}{f(x) \left(g(x)^2-1\right) g'(x)}-\frac{y(x) \left(\left(g(x)^2-1\right) \left(f'(x) \left(2 f'(x) g'(x)+f(x) g''(x)\right)-f(x) f''(x) g'(x)\right)-f(x) g'(x)^2 \left(2 g(x) f'(x)+v (v+1) f(x) g'(x)\right)\right)}{f(x)^2 \left(g(x)^2-1\right) g'(x)}