8.65   ODE No. 1655

$$\boxed{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)-ay\left(x\right){({\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2+1)}^{3/2}=0}}$$

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

Mathematica: cpu = 0.887113 (sec), leaf count = 350 $$\{\{y(x)\to \text{InverseFunction}[-\frac{\sqrt{\frac{{\mathrm{\#}\text{1}}^{2}a+2c_1-2}{c_1-1}}\sqrt{\frac{{\mathrm{\#}\text{1}}^{2}a+2c_1+2}{c_1+1}}(F(i{\sinh }^{-1}\left(\sqrt{\frac{a}{2c_1+2}}\mathrm{\#}\text{1}\right)|\frac{c_1+1}{c_1-1})+(c_1-1)E(i{\sinh }^{-1}\left(\sqrt{\frac{a}{2c_1+2}}\mathrm{\#}\text{1}\right)|\frac{c_1+1}{c_1-1}))}{\sqrt{\frac{a}{2c_1+2}}\sqrt{{\mathrm{\#}\text{1}}^4{a}^2+4{\mathrm{\#}\text{1}}^{2}a{c}_1+4c_1^2-4}}\mathrm{\&}][c_2+x]\},\{y(x)\to \text{InverseFunction}\left[\frac{\sqrt{\frac{{\mathrm{\#}\text{1}}^{2}a+2c_1-2}{c_1-1}}\sqrt{\frac{{\mathrm{\#}\text{1}}^{2}a+2c_1+2}{c_1+1}}(F(i{\sinh }^{-1}\left(\sqrt{\frac{a}{2c_1+2}}\mathrm{\#}\text{1}\right)|\frac{c_1+1}{c_1-1})+(c_1-1)E(i{\sinh }^{-1}\left(\sqrt{\frac{a}{2c_1+2}}\mathrm{\#}\text{1}\right)|\frac{c_1+1}{c_1-1}))}{\sqrt{\frac{a}{2c_1+2}}\sqrt{{\mathrm{\#}\text{1}}^4{a}^2+4{\mathrm{\#}\text{1}}^{2}a{c}_1+4c_1^2-4}}\mathrm{\&}\right][c_2+x]\}\}$$

Maple: cpu = 0.140 (sec), leaf count = 106 $$\{{\int }^{y\left(x\right)}a({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt{-{\mathit{\_}\mathit{a}}^4{a}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{a}}^2{a}^2-4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2+4}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-a({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt{-{\mathit{\_}\mathit{a}}^4{a}^2-4\mathit{\_}\mathit{C}\mathit{1}{\mathit{\_}\mathit{a}}^2{a}^2-4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2+4}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$