8.166   ODE No. 1756

$$\boxed{{\displaystyle ay\left(x\right)\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)+b{\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2-\frac{y\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)}{\sqrt{c^2+x^2}}=0}}$$

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

Mathematica: cpu = 0.662084 (sec), leaf count = 126 $$\left\{\{y(x)\to c_2(-a^2(x{(\sqrt{c^2+x^2}+x)}^{\frac{1}{a}}+c_1)+a{(\sqrt{c^2+x^2}+x)}^{\frac{1}{a}}(\sqrt{c^2+x^2}-bx)+b\sqrt{c^2+x^2}{(\sqrt{c^2+x^2}+x)}^{\frac{1}{a}}+c_1){}^{\frac{a(a^2-1)}{(a-1)(a+1)(a+b)}}\}\right\}$$

Maple: cpu = 3.074 (sec), leaf count = 79 $$\{y\left(x\right)={\left({\left(\frac{a}{a+b}{(-\mathit{\_}\mathit{C}\mathit{1}\sqrt[a]{2}{x}^{a^{-1}+1}{}_2\text{F}{}_1(-\frac{1}{2a},-\frac{1}{2a}-\frac{1}{2};1-a^{-1};-\frac{c^2}{x^2}){(-a^{-1}-1)}^{-1}+\mathit{\_}\mathit{C}\mathit{2})}^{-1}\right)}^{\frac{a}{a+b}}\right)}^{-1}\}$$