4.445   ODE No. 1445

$$\boxed{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)=-\frac{(2f\left(x\right){\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^{2}g\left(x\right)-({\left(g\left(x\right)\right)}^2-1)(f\left(x\right)\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}g\left(x\right)+2\left(\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)))\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)}{f\left(x\right)\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)({\left(g\left(x\right)\right)}^2-1)}-\frac{(({\left(g\left(x\right)\right)}^2-1)(\left(\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)\right)(f\left(x\right)\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}g\left(x\right)+2\left(\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right))-f\left(x\right)\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}f\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right))-(2\left(\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)\right)g\left(x\right)+v(v+1)f\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right))f\left(x\right){\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^2)y\left(x\right)}{{\left(f\left(x\right)\right)}^2\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)({\left(g\left(x\right)\right)}^2-1)}=0}}$$

1. Problem in Latex
2. Mathematica input
3. Maple input

Mathematica: cpu = 1.326669 (sec), leaf count = 169 $$\text{DSolve}[y^{″}(x)=-\frac{y^{\prime }(x)(2f(x)g(x)g^{\prime }(x{)}^2-(g(x{)}^2-1)(2f^{\prime }(x)g^{\prime }(x)+f(x)g^{″}(x)))}{f(x)(g(x{)}^2-1)g^{\prime }(x)}-\frac{y(x)((g(x{)}^2-1)(f^{\prime }(x)(2f^{\prime }(x)g^{\prime }(x)+f(x)g^{″}(x))-f(x)f^{″}(x)g^{\prime }(x))-f(x)g^{\prime }(x{)}^2(2g(x)f^{\prime }(x)+v(v+1)f(x)g^{\prime }(x)))}{f(x{)}^2(g(x{)}^2-1)g^{\prime }(x)},y(x),x]$$

Maple: cpu = 0.203 (sec), leaf count = 21 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}\mathit{L}\mathit{e}\mathit{g}\mathit{e}\mathit{n}\mathit{d}\mathit{r}\mathit{e}\mathit{P}(v,g\left(x\right))f\left(x\right)+\mathit{\_}\mathit{C}\mathit{2}\mathit{L}\mathit{e}\mathit{g}\mathit{e}\mathit{n}\mathit{d}\mathit{r}\mathit{e}\mathit{Q}(v,g\left(x\right))f\left(x\right)\}$$