3.591   ODE No. 591

$$\boxed{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)=\frac{x}{\sqrt{a}y\left(x\right)}F\left(\frac{a{\left(y\left(x\right)\right)}^2+b{x}^2}{a}\right)=0}}$$

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

Mathematica: cpu = 18.189310 (sec), leaf count = 250 $$\text{Solve}[{\int }_1^{y(x)}(-{\int }_1^x(\frac{2bK[1]K[2]F^{\prime }\left(\frac{aK[2{]}^2+bK[1{]}^2}{a}\right)}{\sqrt{a}(\sqrt{a}F\left(\frac{aK[2{]}^2+bK[1{]}^2}{a}\right)+b)}-\frac{2bK[1]K[2]F\left(\frac{aK[2{]}^2+bK[1{]}^2}{a}\right)F^{\prime }\left(\frac{aK[2{]}^2+bK[1{]}^2}{a}\right)}{{(\sqrt{a}F\left(\frac{aK[2{]}^2+bK[1{]}^2}{a}\right)+b)}^2})dK[1]-\frac{bK[2]}{\sqrt{a}F\left(\frac{aK[2{]}^2+b{x}^2}{a}\right)+b})dK[2]+{\int }_1^x\frac{bK[1]F\left(\frac{bK[1{]}^2+ay(x{)}^2}{a}\right)}{\sqrt{a}(\sqrt{a}F\left(\frac{bK[1{]}^2+ay(x{)}^2}{a}\right)+b)}dK[1]=c_1,y(x)]$$

Maple: cpu = 0.140 (sec), leaf count = 108 $$\{y\left(x\right)=\frac{1}{a}\sqrt{a(-b{x}^2+\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}({\int }^{\mathit{\_}\mathit{Z}}{(\sqrt{a}b+F\left(\mathit{\_}\mathit{a}\right)a)}^{-1}d\mathit{\_}\mathit{a}{a}^{\frac{3}{2}}b-b{x}^2+2\mathit{\_}\mathit{C}\mathit{1}a)a)},y\left(x\right)=-\frac{1}{a}\sqrt{a(-b{x}^2+\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}({\int }^{\mathit{\_}\mathit{Z}}{(\sqrt{a}b+F\left(\mathit{\_}\mathit{a}\right)a)}^{-1}d\mathit{\_}\mathit{a}{a}^{\frac{3}{2}}b-b{x}^2+2\mathit{\_}\mathit{C}\mathit{1}a)a)}\}$$