3.560   ODE No. 560

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

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

Mathematica: cpu = 21.646749 (sec), leaf count = 110 $$\{\{y(x)\to -\frac{\sqrt{a^2{c}_1^2(-x^2)-4a^2{c}_{1}x-4a^2+4x^2}}{\sqrt{a^2{c}_1^2-4}}\},\{y(x)\to \frac{\sqrt{a^2{c}_1^2(-x^2)-4a^2{c}_{1}x-4a^2+4x^2}}{\sqrt{a^2{c}_1^2-4}}\}\}$$

Maple: cpu = 0.671 (sec), leaf count = 1512 $$\{{\int }_{\mathit{\_}\mathit{b}}^{x}1(2{\mathit{\_}\mathit{a}}^3-2\mathit{\_}\mathit{a}{\left(y\left(x\right)\right)}^2+\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\left(y\left(x\right)\right)}^2-a^2{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^4)}){(-2a^2\mathit{\_}\mathit{a}{\left(y\left(x\right)\right)}^2+2{\mathit{\_}\mathit{a}}^5+4{\mathit{\_}\mathit{a}}^3{\left(y\left(x\right)\right)}^2+2{\left(y\left(x\right)\right)}^4\mathit{\_}\mathit{a}+{\mathit{\_}\mathit{a}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\left(y\left(x\right)\right)}^2-a^2{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^4)}-{\left(y\left(x\right)\right)}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\left(y\left(x\right)\right)}^2-a^2{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^4)})}^{-1}\mathrm{d}\mathit{\_}\mathit{a}+{\int }^{y\left(x\right)}-(a^2-4x^2)\mathit{\_}\mathit{f}{(-2a^{2}x{\mathit{\_}\mathit{f}}^2+2x^5+4x^3{\mathit{\_}\mathit{f}}^2+2x{\mathit{\_}\mathit{f}}^4+x^2\sqrt{-a^2(-{\mathit{\_}\mathit{f}}^4+{\mathit{\_}\mathit{f}}^2{a}^2-2{\mathit{\_}\mathit{f}}^2{x}^2-x^4)}-{\mathit{\_}\mathit{f}}^2\sqrt{-a^2(-{\mathit{\_}\mathit{f}}^4+{\mathit{\_}\mathit{f}}^2{a}^2-2{\mathit{\_}\mathit{f}}^2{x}^2-x^4)})}^{-1}-{\int }_{\mathit{\_}\mathit{b}}^{x}1(-4\mathit{\_}\mathit{f}\mathit{\_}\mathit{a}-\frac{a^2(-4{\mathit{\_}\mathit{a}}^2\mathit{\_}\mathit{f}-4{\mathit{\_}\mathit{f}}^3+2\mathit{\_}\mathit{f}{a}^2)}{2}\frac{1}{\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}}){(-2a^2\mathit{\_}\mathit{a}{\mathit{\_}\mathit{f}}^2+2{\mathit{\_}\mathit{a}}^5+4{\mathit{\_}\mathit{a}}^3{\mathit{\_}\mathit{f}}^2+2{\mathit{\_}\mathit{f}}^4\mathit{\_}\mathit{a}+{\mathit{\_}\mathit{a}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}-{\mathit{\_}\mathit{f}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)})}^{-1}-1(2{\mathit{\_}\mathit{a}}^3-2\mathit{\_}\mathit{a}{\mathit{\_}\mathit{f}}^2+\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)})(-4a^2\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}+8{\mathit{\_}\mathit{a}}^3\mathit{\_}\mathit{f}+8\mathit{\_}\mathit{a}{\mathit{\_}\mathit{f}}^3-\frac{{\mathit{\_}\mathit{a}}^2{a}^2(-4{\mathit{\_}\mathit{a}}^2\mathit{\_}\mathit{f}-4{\mathit{\_}\mathit{f}}^3+2\mathit{\_}\mathit{f}{a}^2)}{2}\frac{1}{\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}}-2\mathit{\_}\mathit{f}\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}+\frac{{\mathit{\_}\mathit{f}}^2{a}^2(-4{\mathit{\_}\mathit{a}}^2\mathit{\_}\mathit{f}-4{\mathit{\_}\mathit{f}}^3+2\mathit{\_}\mathit{f}{a}^2)}{2}\frac{1}{\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}}){(-2a^2\mathit{\_}\mathit{a}{\mathit{\_}\mathit{f}}^2+2{\mathit{\_}\mathit{a}}^5+4{\mathit{\_}\mathit{a}}^3{\mathit{\_}\mathit{f}}^2+2{\mathit{\_}\mathit{f}}^4\mathit{\_}\mathit{a}+{\mathit{\_}\mathit{a}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}-{\mathit{\_}\mathit{f}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)})}^{-2}\mathrm{d}\mathit{\_}\mathit{a}d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1}=0,{\int }_{\mathit{\_}\mathit{b}}^{x}1(2\mathit{\_}\mathit{a}{\left(y\left(x\right)\right)}^2-2{\mathit{\_}\mathit{a}}^3+\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\left(y\left(x\right)\right)}^2-a^2{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^4)}){(2a^2\mathit{\_}\mathit{a}{\left(y\left(x\right)\right)}^2-2{\mathit{\_}\mathit{a}}^5-4{\mathit{\_}\mathit{a}}^3{\left(y\left(x\right)\right)}^2-2{\left(y\left(x\right)\right)}^4\mathit{\_}\mathit{a}+{\mathit{\_}\mathit{a}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\left(y\left(x\right)\right)}^2-a^2{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^4)}-{\left(y\left(x\right)\right)}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\left(y\left(x\right)\right)}^2-a^2{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^4)})}^{-1}\mathrm{d}\mathit{\_}\mathit{a}+{\int }^{y\left(x\right)}(a^2-4x^2)\mathit{\_}\mathit{f}{(2a^{2}x{\mathit{\_}\mathit{f}}^2-2x^5-4x^3{\mathit{\_}\mathit{f}}^2-2x{\mathit{\_}\mathit{f}}^4+x^2\sqrt{-a^2(-{\mathit{\_}\mathit{f}}^4+{\mathit{\_}\mathit{f}}^2{a}^2-2{\mathit{\_}\mathit{f}}^2{x}^2-x^4)}-{\mathit{\_}\mathit{f}}^2\sqrt{-a^2(-{\mathit{\_}\mathit{f}}^4+{\mathit{\_}\mathit{f}}^2{a}^2-2{\mathit{\_}\mathit{f}}^2{x}^2-x^4)})}^{-1}-{\int }_{\mathit{\_}\mathit{b}}^x-1(2\mathit{\_}\mathit{a}{\mathit{\_}\mathit{f}}^2-2{\mathit{\_}\mathit{a}}^3+\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)})(4a^2\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}-8{\mathit{\_}\mathit{a}}^3\mathit{\_}\mathit{f}-8\mathit{\_}\mathit{a}{\mathit{\_}\mathit{f}}^3-\frac{{\mathit{\_}\mathit{a}}^2{a}^2(-4{\mathit{\_}\mathit{a}}^2\mathit{\_}\mathit{f}-4{\mathit{\_}\mathit{f}}^3+2\mathit{\_}\mathit{f}{a}^2)}{2}\frac{1}{\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}}-2\mathit{\_}\mathit{f}\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}+\frac{{\mathit{\_}\mathit{f}}^2{a}^2(-4{\mathit{\_}\mathit{a}}^2\mathit{\_}\mathit{f}-4{\mathit{\_}\mathit{f}}^3+2\mathit{\_}\mathit{f}{a}^2)}{2}\frac{1}{\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}}){(2a^2\mathit{\_}\mathit{a}{\mathit{\_}\mathit{f}}^2-2{\mathit{\_}\mathit{a}}^5-4{\mathit{\_}\mathit{a}}^3{\mathit{\_}\mathit{f}}^2-2{\mathit{\_}\mathit{f}}^4\mathit{\_}\mathit{a}+{\mathit{\_}\mathit{a}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}-{\mathit{\_}\mathit{f}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)})}^{-2}+1(4\mathit{\_}\mathit{f}\mathit{\_}\mathit{a}-\frac{a^2(-4{\mathit{\_}\mathit{a}}^2\mathit{\_}\mathit{f}-4{\mathit{\_}\mathit{f}}^3+2\mathit{\_}\mathit{f}{a}^2)}{2}\frac{1}{\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}}){(2a^2\mathit{\_}\mathit{a}{\mathit{\_}\mathit{f}}^2-2{\mathit{\_}\mathit{a}}^5-4{\mathit{\_}\mathit{a}}^3{\mathit{\_}\mathit{f}}^2-2{\mathit{\_}\mathit{f}}^4\mathit{\_}\mathit{a}+{\mathit{\_}\mathit{a}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)}-{\mathit{\_}\mathit{f}}^2\sqrt{a^2({\mathit{\_}\mathit{a}}^4+2{\mathit{\_}\mathit{a}}^2{\mathit{\_}\mathit{f}}^2+{\mathit{\_}\mathit{f}}^4-{\mathit{\_}\mathit{f}}^2{a}^2)})}^{-1}\mathrm{d}\mathit{\_}\mathit{a}d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$