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 = 0.334332 (sec), leaf count = 126

$$\{\{y(x)\to -\frac{\sqrt{2(a^2-1)x{e}^{(a^2-1)c_1}+e^{2(a^2-1)c_1}+{(a^2-1)}^3(-x^2)}}{\sqrt{{(a^2-1)}^3}}\},\{y(x)\to \frac{\sqrt{2(a^2-1)x{e}^{(a^2-1)c_1}+e^{2(a^2-1)c_1}+{(a^2-1)}^3(-x^2)}}{\sqrt{{(a^2-1)}^3}}\}\}$$

✓ Maple : cpu = 1.421 (sec), leaf count = 215

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