3.380   ODE No. 380

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

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

Mathematica: cpu = 0.410552 (sec), leaf count = 1757 $$\{\{y(x)\to -\frac{x^2}{4}-\frac{1}{4}\sqrt[3]{x^6-20\cosh \left(3c_1\right)x^3-20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{-\cosh \left(3c_1\right)x^9-\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6-3\cosh \left(9c_1\right)x^3-3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}+\frac{-9x^4-72\cosh \left(3c_1\right)x-72\sinh \left(3c_1\right)x}{36\sqrt[3]{x^6-20\cosh \left(3c_1\right)x^3-20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{-\cosh \left(3c_1\right)x^9-\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6-3\cosh \left(9c_1\right)x^3-3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}}\},\{y(x)\to -\frac{x^2}{4}+\frac{1}{8}(1-i\sqrt{3})\sqrt[3]{x^6-20\cosh \left(3c_1\right)x^3-20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{-\cosh \left(3c_1\right)x^9-\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6-3\cosh \left(9c_1\right)x^3-3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}-\frac{(1+i\sqrt{3})(-9x^4-72\cosh \left(3c_1\right)x-72\sinh \left(3c_1\right)x)}{72\sqrt[3]{x^6-20\cosh \left(3c_1\right)x^3-20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{-\cosh \left(3c_1\right)x^9-\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6-3\cosh \left(9c_1\right)x^3-3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}}\},\{y(x)\to -\frac{x^2}{4}+\frac{1}{8}(1+i\sqrt{3})\sqrt[3]{x^6-20\cosh \left(3c_1\right)x^3-20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{-\cosh \left(3c_1\right)x^9-\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6-3\cosh \left(9c_1\right)x^3-3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}-\frac{(1-i\sqrt{3})(-9x^4-72\cosh \left(3c_1\right)x-72\sinh \left(3c_1\right)x)}{72\sqrt[3]{x^6-20\cosh \left(3c_1\right)x^3-20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{-\cosh \left(3c_1\right)x^9-\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6-3\cosh \left(9c_1\right)x^3-3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}}\},\{y(x)\to -\frac{x^2}{4}-\frac{1}{4}\sqrt[3]{x^6+20\cosh \left(3c_1\right)x^3+20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{\cosh \left(3c_1\right)x^9+\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6+3\cosh \left(9c_1\right)x^3+3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}+\frac{-9x^4+72\cosh \left(3c_1\right)x+72\sinh \left(3c_1\right)x}{36\sqrt[3]{x^6+20\cosh \left(3c_1\right)x^3+20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{\cosh \left(3c_1\right)x^9+\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6+3\cosh \left(9c_1\right)x^3+3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}}\},\{y(x)\to -\frac{x^2}{4}+\frac{1}{8}(1-i\sqrt{3})\sqrt[3]{x^6+20\cosh \left(3c_1\right)x^3+20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{\cosh \left(3c_1\right)x^9+\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6+3\cosh \left(9c_1\right)x^3+3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}-\frac{(1+i\sqrt{3})(-9x^4+72\cosh \left(3c_1\right)x+72\sinh \left(3c_1\right)x)}{72\sqrt[3]{x^6+20\cosh \left(3c_1\right)x^3+20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{\cosh \left(3c_1\right)x^9+\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6+3\cosh \left(9c_1\right)x^3+3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}}\},\{y(x)\to -\frac{x^2}{4}+\frac{1}{8}(1+i\sqrt{3})\sqrt[3]{x^6+20\cosh \left(3c_1\right)x^3+20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{\cosh \left(3c_1\right)x^9+\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6+3\cosh \left(9c_1\right)x^3+3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}-\frac{(1-i\sqrt{3})(-9x^4+72\cosh \left(3c_1\right)x+72\sinh \left(3c_1\right)x)}{72\sqrt[3]{x^6+20\cosh \left(3c_1\right)x^3+20\sinh \left(3c_1\right)x^3-8\cosh \left(6c_1\right)-8\sinh \left(6c_1\right)+8\sqrt{\cosh \left(3c_1\right)x^9+\sinh \left(3c_1\right)x^9+3\cosh \left(6c_1\right)x^6+3\sinh \left(6c_1\right)x^6+3\cosh \left(9c_1\right)x^3+3\sinh \left(9c_1\right)x^3+\cosh \left(12c_1\right)+\sinh \left(12c_1\right)}}}\}\}$$

Maple: cpu = 0.437 (sec), leaf count = 690 $$\{y\left(x\right)={(\frac{1}{2}\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}+\frac{x^2}{2}\frac{1}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}-\frac{x}{2})}^2+2x(1/2\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}+1/2\frac{x^2}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}-x/2),y\left(x\right)={(-\frac{1}{4}\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-\frac{x^2}{4}\frac{1}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}-\frac{x}{2}-\frac{i}{2}\sqrt{3}(\frac{1}{2}\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-\frac{x^2}{2}\frac{1}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}))}^2+2x(-1/4\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-1/4\frac{x^2}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}-x/2-i/2\sqrt{3}(1/2\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-1/2\frac{x^2}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}})),y\left(x\right)={(-\frac{1}{4}\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-\frac{x^2}{4}\frac{1}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}-\frac{x}{2}+\frac{i}{2}\sqrt{3}(\frac{1}{2}\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-\frac{x^2}{2}\frac{1}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}))}^2+2x(-1/4\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-1/4\frac{x^2}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}-x/2+i/2\sqrt{3}(1/2\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-1/2\frac{x^2}{\sqrt[3]{6\mathit{\_}\mathit{C}\mathit{1}-x^3+2\sqrt{-3\mathit{\_}\mathit{C}\mathit{1}{x}^3+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}))\}$$