2.1655   ODE No. 1655

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

$$y^{″}(x)-ay(x){(y^{\prime }(x{)}^2+1)}^{3/2}=0$$ ✓ Mathematica : cpu = 0.800467 (sec), leaf count = 350

$$\{\{y(x)\to \text{InverseFunction}[-\frac{\sqrt{\frac{{\mathrm{\#}\text{1}}^{2}a-2+2c_1}{-1+c_1}}\sqrt{\frac{{\mathrm{\#}\text{1}}^{2}a+2+2c_1}{1+c_1}}(F(i{\sinh }^{-1}\left(\sqrt{\frac{a}{2c_1+2}}\mathrm{\#}\text{1}\right)|\frac{c_1+1}{c_1-1})+(-1+c_1)E(i{\sinh }^{-1}\left(\sqrt{\frac{a}{2c_1+2}}\mathrm{\#}\text{1}\right)|\frac{c_1+1}{c_1-1}))}{\sqrt{\frac{a}{2+2c_1}}\sqrt{{\mathrm{\#}\text{1}}^4{a}^2+4{\mathrm{\#}\text{1}}^{2}a{c}_1-4+4c_1{}^2}}\mathrm{\&}][x+c_2]\},\{y(x)\to \text{InverseFunction}\left[\frac{\sqrt{\frac{{\mathrm{\#}\text{1}}^{2}a-2+2c_1}{-1+c_1}}\sqrt{\frac{{\mathrm{\#}\text{1}}^{2}a+2+2c_1}{1+c_1}}(F(i{\sinh }^{-1}\left(\sqrt{\frac{a}{2c_1+2}}\mathrm{\#}\text{1}\right)|\frac{c_1+1}{c_1-1})+(-1+c_1)E(i{\sinh }^{-1}\left(\sqrt{\frac{a}{2c_1+2}}\mathrm{\#}\text{1}\right)|\frac{c_1+1}{c_1-1}))}{\sqrt{\frac{a}{2+2c_1}}\sqrt{{\mathrm{\#}\text{1}}^4{a}^2+4{\mathrm{\#}\text{1}}^{2}a{c}_1-4+4c_1{}^2}}\mathrm{\&}\right][x+c_2]\}\}$$ ✓ Maple : cpu = 0.476 (sec), leaf count = 84

$$\{{\int }^{y\left(x\right)}a({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt{4-a^2{({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})}^2}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-a({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt{4-a^2{({\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{C}\mathit{1})}^2}}d\mathit{\_}\mathit{a}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$