8.96   ODE No. 1686

$$\boxed{{\displaystyle x^4\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)+a^2{\left(y\left(x\right)\right)}^n=0}}$$

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

Mathematica: cpu = 0.032504 (sec), leaf count = 23 $$\text{DSolve}[a^{2}y(x{)}^n+x^4{y}^{″}(x)=0,y(x),x]$$

Maple: cpu = 0.764 (sec), leaf count = 160 $$\{y\left(x\right)=\mathit{O}\mathit{D}\mathit{E}\mathit{S}\mathit{o}\mathit{l}\mathit{S}\mathit{t}\mathit{r}\mathit{u}\mathit{c}(\mathit{\_}\mathit{a}{\mathrm{e}}^{\int \mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\mathrm{d}\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}},[\{\frac{\mathrm{d}}{\mathrm{d}\mathit{\_}\mathit{a}}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)=(\frac{{\mathit{\_}\mathit{a}}^n{n}^2{a}^2}{4}-\frac{{\mathit{\_}\mathit{a}}^{n}n{a}^2}{2}+\frac{{\mathit{\_}\mathit{a}}^n{a}^2}{4}-\frac{\mathit{\_}\mathit{a}n}{2}+\frac{3\mathit{\_}\mathit{a}}{2}){\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^3+(-\frac{n}{2}+\frac{5}{2}){\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^2\},\{\mathit{\_}\mathit{a}=y\left(x\right)x^{-2{(n-1)}^{-1}},\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)=-2\frac{1}{-nx\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+x\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+2y\left(x\right)}{\left(x^{-2{(n-1)}^{-1}}\right)}^{-1}\},\{x={\mathrm{e}}^{\frac{(\int \mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\mathrm{d}\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})n}{2}-\frac{\int \mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\mathrm{d}\mathit{\_}\mathit{a}}{2}-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2}},y\left(x\right)=\mathit{\_}\mathit{a}{\mathrm{e}}^{\int \mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\mathrm{d}\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}}\}])\}$$