3.549   ODE No. 549

$$\boxed{{\displaystyle x^2{({\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2+1)}^3-a^2=0}}$$

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

Mathematica: cpu = 0.246031 (sec), leaf count = 406 $$\{\{y(x)\to c_1-\frac{\sqrt[3]{x}(2x^{2/3}+(1+i\sqrt{3})a^{2/3})\sqrt{\frac{-2x^{2/3}+(-1-i\sqrt{3})a^{2/3}}{x^{2/3}}}}{2\sqrt{2}}\},\{y(x)\to c_1+\frac{\sqrt[3]{x}\sqrt{\frac{-2x^{2/3}+(-1-i\sqrt{3})a^{2/3}}{x^{2/3}}}(2x^{2/3}+(1+i\sqrt{3})a^{2/3})}{2\sqrt{2}}\},\{y(x)\to c_1-\frac{\sqrt[3]{x}(2x^{2/3}+(1-i\sqrt{3})a^{2/3})\sqrt{\frac{-2x^{2/3}+i(\sqrt{3}+i)a^{2/3}}{x^{2/3}}}}{2\sqrt{2}}\},\{y(x)\to c_1+\frac{\sqrt[3]{x}\sqrt{\frac{-2x^{2/3}+i(\sqrt{3}+i)a^{2/3}}{x^{2/3}}}(2x^{2/3}+(1-i\sqrt{3})a^{2/3})}{2\sqrt{2}}\},\{y(x)\to c_1-\sqrt{\frac{a^{2/3}}{x^{2/3}}-1}(x-a^{2/3}\sqrt[3]{x})\},\{y(x)\to \sqrt{\frac{a^{2/3}}{x^{2/3}}-1}(x-a^{2/3}\sqrt[3]{x})+c_1\}\}$$

Maple: cpu = 0.140 (sec), leaf count = 552 $$\{y\left(x\right)=-1\sqrt{-\frac{1}{a^4}{\left(a^{2}x\right)}^{\frac{4}{3}}({\left(a^{2}x\right)}^{\frac{2}{3}}-a^2)}({\left(a^{2}x\right)}^{\frac{2}{3}}-a^2){\left(a^{2}x\right)}^{-\frac{2}{3}}+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=1\sqrt{-\frac{1}{a^4}{\left(a^{2}x\right)}^{\frac{4}{3}}({\left(a^{2}x\right)}^{\frac{2}{3}}-a^2)}({\left(a^{2}x\right)}^{\frac{2}{3}}-a^2){\left(a^{2}x\right)}^{-\frac{2}{3}}+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=-\frac{i}{4}\sqrt{2}\sqrt{\frac{i}{a^4}{\left(a^{2}x\right)}^{\frac{4}{3}}(\sqrt{3}{a}^2+2i{\left(a^{2}x\right)}^{\frac{2}{3}}+i{a}^2)}(\sqrt{3}{a}^2+2i{\left(a^{2}x\right)}^{\frac{2}{3}}+i{a}^2){\left(a^{2}x\right)}^{-\frac{2}{3}}+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=\frac{i}{4}\sqrt{2}\sqrt{\frac{i}{a^4}{\left(a^{2}x\right)}^{\frac{4}{3}}(\sqrt{3}{a}^2+2i{\left(a^{2}x\right)}^{\frac{2}{3}}+i{a}^2)}(\sqrt{3}{a}^2+2i{\left(a^{2}x\right)}^{\frac{2}{3}}+i{a}^2){\left(a^{2}x\right)}^{-\frac{2}{3}}+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=-\frac{i}{4}\sqrt{2}\sqrt{-i(\sqrt{3}\sqrt[3]{a^{2}x}-i\sqrt[3]{a^{2}x}-2ix)x}\sqrt{\frac{1}{a^4}{\left(a^{2}x\right)}^{\frac{4}{3}}(\sqrt{3}{a}^2-2i{\left(a^{2}x\right)}^{\frac{2}{3}}-i{a}^2)}(\sqrt{3}{a}^2-2i{\left(a^{2}x\right)}^{\frac{2}{3}}-i{a}^2)\frac{1}{\sqrt{(\sqrt{3}\sqrt[3]{a^{2}x}-i\sqrt[3]{a^{2}x}-2ix)x}}{\left(a^{2}x\right)}^{-\frac{2}{3}}+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=\frac{i}{4}\sqrt{2}\sqrt{-i(\sqrt{3}\sqrt[3]{a^{2}x}-i\sqrt[3]{a^{2}x}-2ix)x}\sqrt{\frac{1}{a^4}{\left(a^{2}x\right)}^{\frac{4}{3}}(\sqrt{3}{a}^2-2i{\left(a^{2}x\right)}^{\frac{2}{3}}-i{a}^2)}(\sqrt{3}{a}^2-2i{\left(a^{2}x\right)}^{\frac{2}{3}}-i{a}^2)\frac{1}{\sqrt{(\sqrt{3}\sqrt[3]{a^{2}x}-i\sqrt[3]{a^{2}x}-2ix)x}}{\left(a^{2}x\right)}^{-\frac{2}{3}}+\mathit{\_}\mathit{C}\mathit{1}\}$$