8.234   ODE No. 1824

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

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

Mathematica: cpu = 0.352545 (sec), leaf count = 347 $$\{\{y(x)\to \frac{-2\sqrt{x^2(a^2+c_1^2-x^2)}+c_{1}x\log (-c_1(\sqrt{x^2(a^2+c_1^2-x^2)}+c_{1}x)+a^2(-x)+a{x}^2)+c_{1}x\log (c_1(\sqrt{x^2(a^2+c_1^2-x^2)}+c_{1}x)+a^{2}x+a{x}^2)+c_{1}x\log (x-a)-c_{1}x\log (x(x-a))-c_{1}x\log (a+x)-c_{1}x\log (x(a+x))}{2x}+c_2\},\{y(x)\to c_2-\frac{-2\sqrt{x^2(a^2+c_1^2-x^2)}+c_{1}x\log (-c_1(\sqrt{x^2(a^2+c_1^2-x^2)}+c_{1}x)+a^2(-x)+a{x}^2)+c_{1}x\log (c_1(\sqrt{x^2(a^2+c_1^2-x^2)}+c_{1}x)+a^{2}x+a{x}^2)+c_1(-x)\log (x-a)-c_{1}x\log (x(x-a))+c_{1}x\log (a+x)-c_{1}x\log (x(a+x))}{2x}\}\}$$

Maple: cpu = 0.717 (sec), leaf count = 99 $$\{y\left(x\right)=\int \frac{1}{a(a^2-x^2)}(-\mathit{\_}\mathit{C}\mathit{1}{a}^2+x\sqrt{a^2({\mathit{\_}\mathit{C}\mathit{1}}^2+a^2-x^2)})\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{2},y\left(x\right)=\int -\frac{1}{a(a^2-x^2)}(\mathit{\_}\mathit{C}\mathit{1}{a}^2+x\sqrt{a^2({\mathit{\_}\mathit{C}\mathit{1}}^2+a^2-x^2)})\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{2}\}$$