3.499   ODE No. 499

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

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

Mathematica: cpu = 0.333542 (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 = 0.702 (sec), leaf count = 201 $$\{y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-\ln \left(x\right)+{\int }^{\mathit{\_}\mathit{Z}}\frac{\mathit{\_}\mathit{a}}{a^2{\mathit{\_}\mathit{a}}^4-{\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2}(-{\mathit{\_}\mathit{a}}^2{a}^2+{\mathit{\_}\mathit{a}}^2-a^2+\sqrt{{\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2})d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})x,y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-\ln \left(x\right)-{\int }^{\mathit{\_}\mathit{Z}}\frac{\mathit{\_}\mathit{a}}{a^2{\mathit{\_}\mathit{a}}^4-{\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2}({\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2+\sqrt{{\mathit{\_}\mathit{a}}^2{a}^2-{\mathit{\_}\mathit{a}}^2+a^2})d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})x,y\left(x\right)=ax\frac{1}{\sqrt{-a^2+1}},y\left(x\right)=-ax\frac{1}{\sqrt{-a^2+1}}\}$$