2.381   ODE No. 381

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

$$y^{\prime }(x{)}^2-2x{y}^{\prime }(x)+y(x)=0$$ ✓ Mathematica : cpu = 0.440223 (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 = 0.62 (sec), leaf count = 656

$$\{y\left(x\right)=-{(\frac{1}{2}\sqrt[3]{-6\mathit{\_}\mathit{C}\mathit{1}+x^3+2\sqrt{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}+1/2\frac{x^2}{\sqrt[3]{-6\mathit{\_}\mathit{C}\mathit{1}+x^3+2\sqrt{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}))}^2+2x(-1/4\sqrt[3]{-6\mathit{\_}\mathit{C}\mathit{1}+x^3+2\sqrt{-3x^3\mathit{\_}\mathit{C}\mathit{1}+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-1/4\frac{x^2}{\sqrt[3]{-6\mathit{\_}\mathit{C}\mathit{1}+x^3+2\sqrt{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-1/2\frac{x^2}{\sqrt[3]{-6\mathit{\_}\mathit{C}\mathit{1}+x^3+2\sqrt{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}))}^2+2x(-1/4\sqrt[3]{-6\mathit{\_}\mathit{C}\mathit{1}+x^3+2\sqrt{-3x^3\mathit{\_}\mathit{C}\mathit{1}+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-1/4\frac{x^2}{\sqrt[3]{-6\mathit{\_}\mathit{C}\mathit{1}+x^3+2\sqrt{-3x^3\mathit{\_}\mathit{C}\mathit{1}+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{-3x^3\mathit{\_}\mathit{C}\mathit{1}+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}-1/2\frac{x^2}{\sqrt[3]{-6\mathit{\_}\mathit{C}\mathit{1}+x^3+2\sqrt{-3x^3\mathit{\_}\mathit{C}\mathit{1}+9{\mathit{\_}\mathit{C}\mathit{1}}^2}}}))\}$$