8.155   ODE No. 1745

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

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

Mathematica: cpu = 0.294537 (sec), leaf count = 204 $$\{\{y(x)\to \text{InverseFunction}[-\sqrt{2}(\frac{1}{2}\sqrt{a-\mathrm{\#}\text{1}}\sqrt{e^{2c_1}-2(a-\mathrm{\#}\text{1})}-\frac{e^{2c_1}{\tan }^{-1}\left(\frac{\sqrt{2}\sqrt{a-\mathrm{\#}\text{1}}}{\sqrt{e^{2c_1}-2(a-\mathrm{\#}\text{1})}}\right)}{2\sqrt{2}})\mathrm{\&}][c_2+x]\},\{y(x)\to \text{InverseFunction}\left[\sqrt{2}(\frac{1}{2}\sqrt{a-\mathrm{\#}\text{1}}\sqrt{e^{2c_1}-2(a-\mathrm{\#}\text{1})}-\frac{e^{2c_1}{\tan }^{-1}\left(\frac{\sqrt{2}\sqrt{a-\mathrm{\#}\text{1}}}{\sqrt{e^{2c_1}-2(a-\mathrm{\#}\text{1})}}\right)}{2\sqrt{2}})\mathrm{\&}\right][c_2+x]\}\}$$

Maple: cpu = 1.763 (sec), leaf count = 293 $$\{-\sqrt{-{\left(y\left(x\right)\right)}^2+(2a+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)-a(\mathit{\_}\mathit{C}\mathit{1}+a)}+\arctan \left(1(y\left(x\right)-a-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2})\frac{1}{\sqrt{-{\left(y\left(x\right)\right)}^2+(2a+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)-a(\mathit{\_}\mathit{C}\mathit{1}+a)}}\right)a+\frac{\mathit{\_}\mathit{C}\mathit{1}}{2}\arctan \left(1(y\left(x\right)-a-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2})\frac{1}{\sqrt{-{\left(y\left(x\right)\right)}^2+(2a+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)-a(\mathit{\_}\mathit{C}\mathit{1}+a)}}\right)-a\arctan \left(1(y\left(x\right)-a-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2})\frac{1}{\sqrt{-{\left(y\left(x\right)\right)}^2+(2a+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)+a(-\mathit{\_}\mathit{C}\mathit{1}-a)}}\right)-x-\mathit{\_}\mathit{C}\mathit{2}=0,\sqrt{-{\left(y\left(x\right)\right)}^2+(2a+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)-a(\mathit{\_}\mathit{C}\mathit{1}+a)}-\arctan \left(1(y\left(x\right)-a-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2})\frac{1}{\sqrt{-{\left(y\left(x\right)\right)}^2+(2a+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)-a(\mathit{\_}\mathit{C}\mathit{1}+a)}}\right)a-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2}\arctan \left(1(y\left(x\right)-a-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2})\frac{1}{\sqrt{-{\left(y\left(x\right)\right)}^2+(2a+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)-a(\mathit{\_}\mathit{C}\mathit{1}+a)}}\right)+a\arctan \left(1(y\left(x\right)-a-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2})\frac{1}{\sqrt{-{\left(y\left(x\right)\right)}^2+(2a+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)+a(-\mathit{\_}\mathit{C}\mathit{1}-a)}}\right)-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$