3.559   ODE No. 559

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

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

Mathematica: cpu = 0.351045 (sec), leaf count = 212 $$\{\{y(x)\to -\frac{\sqrt{a^6(-x^2)+3a^4{x}^2+2a^{2}x{e}^{a^2{c}_1-c_1}-2x{e}^{a^2{c}_1-c_1}+e^{2a^2{c}_1-2c_1}-3a^2{x}^2+x^2}}{\sqrt{a^6-3a^4+3a^2-1}}\},\{y(x)\to \frac{\sqrt{a^6(-x^2)+3a^4{x}^2+2a^{2}x{e}^{a^2{c}_1-c_1}-2x{e}^{a^2{c}_1-c_1}+e^{2a^2{c}_1-2c_1}-3a^2{x}^2+x^2}}{\sqrt{a^6-3a^4+3a^2-1}}\}\}$$

Maple: cpu = 0.234 (sec), leaf count = 223 $$\{x-{\mathrm{e}}^{{\int }^{\frac{1}{(a^2-1)y\left(x\right)}(-a^{2}x+\sqrt{a^2{x}^2+a^2{\left(y\left(x\right)\right)}^2-{\left(y\left(x\right)\right)}^2})}a(a\sqrt{{\mathit{\_}\mathit{a}}^2+1}-\mathit{\_}\mathit{a})\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}{(-\mathit{\_}\mathit{a}a+\sqrt{{\mathit{\_}\mathit{a}}^2+1})}^{-1}{(-{\mathit{\_}\mathit{a}}^{2}a+\sqrt{{\mathit{\_}\mathit{a}}^2+1}\mathit{\_}\mathit{a}-a)}^{-1}d\mathit{\_}\mathit{a}}\mathit{\_}\mathit{C}\mathit{1}=0,x-{\mathrm{e}}^{{\int }^{-\frac{1}{(a^2-1)y\left(x\right)}(a^{2}x+\sqrt{a^2{x}^2+a^2{\left(y\left(x\right)\right)}^2-{\left(y\left(x\right)\right)}^2})}a(a\sqrt{{\mathit{\_}\mathit{a}}^2+1}-\mathit{\_}\mathit{a})\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+1}}{(-\mathit{\_}\mathit{a}a+\sqrt{{\mathit{\_}\mathit{a}}^2+1})}^{-1}{(-{\mathit{\_}\mathit{a}}^{2}a+\sqrt{{\mathit{\_}\mathit{a}}^2+1}\mathit{\_}\mathit{a}-a)}^{-1}d\mathit{\_}\mathit{a}}\mathit{\_}\mathit{C}\mathit{1}=0\}$$