8.64   ODE No. 1654

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

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

Mathematica: cpu = 0.311540 (sec), leaf count = 308 $$\{\{y(x)\to c_2-\frac{\sqrt{\frac{a{x}^2+c_1-1}{c_1-1}}\sqrt{\frac{a{x}^2+c_1+1}{c_1+1}}(F(i{\sinh }^{-1}\left(x\sqrt{\frac{a}{c_1+1}}\right)|\frac{c_1+1}{c_1-1})+(c_1-1)E(i{\sinh }^{-1}\left(x\sqrt{\frac{a}{c_1+1}}\right)|\frac{c_1+1}{c_1-1}))}{\sqrt{\frac{a}{c_1+1}}\sqrt{a^2{x}^4+2a{c}_1{x}^2+c_1^2-1}}\},\{y(x)\to c_2+\frac{\sqrt{\frac{a{x}^2+c_1-1}{c_1-1}}\sqrt{\frac{a{x}^2+c_1+1}{c_1+1}}(F(i{\sinh }^{-1}\left(x\sqrt{\frac{a}{c_1+1}}\right)|\frac{c_1+1}{c_1-1})+(c_1-1)E(i{\sinh }^{-1}\left(x\sqrt{\frac{a}{c_1+1}}\right)|\frac{c_1+1}{c_1-1}))}{\sqrt{\frac{a}{c_1+1}}\sqrt{a^2{x}^4+2a{c}_1{x}^2+c_1^2-1}}\}\}$$

Maple: cpu = 0.125 (sec), leaf count = 49 $$\{y\left(x\right)=\int \sqrt{-{(a^2{x}^4+4\mathit{\_}\mathit{C}\mathit{1}{a}^2{x}^2+4{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2-1)}^{-1}}a(x^2+2\mathit{\_}\mathit{C}\mathit{1})\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{2}\}$$