2.499   ODE No. 499

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

$$a^2(-x^2)-2a^{2}xy(x)y^{\prime }(x)+(1-a^2)y(x{)}^2{y}^{\prime }(x{)}^2+y(x{)}^2=0$$ ✓ Mathematica : cpu = 0.363784 (sec), leaf count = 212

$$\{\{y(x)\to -\frac{\sqrt{a^6(-x^2)+3a^4{x}^2-3a^2{x}^2+2a^{2}x{e}^{a^2{c}_1-c_1}-2x{e}^{a^2{c}_1-c_1}+e^{2a^2{c}_1-2c_1}+x^2}}{\sqrt{a^6-3a^4+3a^2-1}}\},\{y(x)\to \frac{\sqrt{a^6(-x^2)+3a^4{x}^2-3a^2{x}^2+2a^{2}x{e}^{a^2{c}_1-c_1}-2x{e}^{a^2{c}_1-c_1}+e^{2a^2{c}_1-2c_1}+x^2}}{\sqrt{a^6-3a^4+3a^2-1}}\}\}$$ ✓ Maple : cpu = 0.238 (sec), leaf count = 189

$$\{y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-\ln \left(x\right)+{\int }^{\mathit{\_}\mathit{Z}}\frac{\mathit{\_}\mathit{a}}{({\mathit{\_}\mathit{a}}^2+1)({\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2)}(-{\mathit{\_}\mathit{a}}^2{a}^2+{\mathit{\_}\mathit{a}}^2-a^2+\sqrt{{\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2})d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})x,y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-\ln \left(x\right)-{\int }^{\mathit{\_}\mathit{Z}}\frac{\mathit{\_}\mathit{a}}{({\mathit{\_}\mathit{a}}^2+1)({\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2)}({\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2+\sqrt{{\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2})d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})x,y\left(x\right)=ax\frac{1}{\sqrt{-a^2+1}},y\left(x\right)=-ax\frac{1}{\sqrt{-a^2+1}}\}$$