2.559   ODE No. 559

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

$$-ay(x)y^{\prime }(x)-ax+y(x)\sqrt{y^{\prime }(x{)}^2+1}=0$$ ✓ Mathematica : cpu = 0.308467 (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 = 0.626 (sec), leaf count = 223

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