4.18.20 $x{y}^{\prime }(x{)}^2+2y(x)y^{\prime }(x)-x=0$

ODE
$$x{y}^{\prime }(x{)}^2+2y(x)y^{\prime }(x)-x=0$$ ODE Classification

[[_homogeneous, `class A`], _dAlembert]

Book solution method
Homogeneous ODE, $x^{n}f(\frac{y}{x},y^{\prime })=0$, Solve for $p$

Mathematica ✓
cpu = 1.46566 (sec), leaf count = 6977

$$\{\{y(x)\to \frac{-\sqrt{\frac{36x^6+\frac{362^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}{x}^2+\cosh \left(6c_1\right)+\sinh \left(6c_1\right)}{x^4}}{x}^2-9\sqrt{\frac{8x^2}{9}-\frac{42^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))}{9\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}-\frac{\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}{9x^2}+\frac{2\cosh \left(6c_1\right)}{81x^4}+\frac{2\sinh \left(6c_1\right)}{81x^4}-\frac{2(-432\cosh \left(3c_1\right)x^6-432\sinh \left(3c_1\right)x^6+\cosh \left(9c_1\right)+\sinh \left(9c_1\right))}{81\sqrt{\frac{36x^6+\frac{362^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}{x}^2+\cosh \left(6c_1\right)+\sinh \left(6c_1\right)}{x^4}}{x}^6}}{x}^2+\cosh \left(3c_1\right)+\sinh \left(3c_1\right)}{18x^2}\},\{y(x)\to \frac{-\sqrt{\frac{36x^6+\frac{362^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}{x}^2+\cosh \left(6c_1\right)+\sinh \left(6c_1\right)}{x^4}}{x}^2+9\sqrt{\frac{8x^2}{9}-\frac{42^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))}{9\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}-\frac{\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}{9x^2}+\frac{2\cosh \left(6c_1\right)}{81x^4}+\frac{2\sinh \left(6c_1\right)}{81x^4}-\frac{2(-432\cosh \left(3c_1\right)x^6-432\sinh \left(3c_1\right)x^6+\cosh \left(9c_1\right)+\sinh \left(9c_1\right))}{81\sqrt{\frac{36x^6+\frac{362^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}{x}^2+\cosh \left(6c_1\right)+\sinh \left(6c_1\right)}{x^4}}{x}^6}}{x}^2+\cosh \left(3c_1\right)+\sinh \left(3c_1\right)}{18x^2}\},\{y(x)\to \frac{\sqrt{\frac{36x^6+\frac{362^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}{x}^2+\cosh \left(6c_1\right)+\sinh \left(6c_1\right)}{x^4}}{x}^2-9\sqrt{\frac{8x^2}{9}-\frac{42^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))}{9\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}-\frac{\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}{9x^2}+\frac{2\cosh \left(6c_1\right)}{81x^4}+\frac{2\sinh \left(6c_1\right)}{81x^4}+\frac{2(-432\cosh \left(3c_1\right)x^6-432\sinh \left(3c_1\right)x^6+\cosh \left(9c_1\right)+\sinh \left(9c_1\right))}{81\sqrt{\frac{36x^6+\frac{362^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}{x}^2+\cosh \left(6c_1\right)+\sinh \left(6c_1\right)}{x^4}}{x}^6}}{x}^2+\cosh \left(3c_1\right)+\sinh \left(3c_1\right)}{18x^2}\},\{y(x)\to \frac{\sqrt{\frac{36x^6+\frac{362^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}{x}^2+\cosh \left(6c_1\right)+\sinh \left(6c_1\right)}{x^4}}{x}^2+9\sqrt{\frac{8x^2}{9}-\frac{42^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))}{9\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}-\frac{\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}{9x^2}+\frac{2\cosh \left(6c_1\right)}{81x^4}+\frac{2\sinh \left(6c_1\right)}{81x^4}+\frac{2(-432\cosh \left(3c_1\right)x^6-432\sinh \left(3c_1\right)x^6+\cosh \left(9c_1\right)+\sinh \left(9c_1\right))}{81\sqrt{\frac{36x^6+\frac{362^{2/3}(2x^6-\cosh \left(6c_1\right)-\sinh \left(6c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6+40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)-\sinh \left(12c_1\right)+\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(1-16x^6)\sinh \left(3c_1\right)){}^3(\cosh \left(15c_1\right)+\sinh \left(15c_1\right))}}{x}^2+\cosh \left(6c_1\right)+\sinh \left(6c_1\right)}{x^4}}{x}^6}}{x}^2+\cosh \left(3c_1\right)+\sinh \left(3c_1\right)}{18x^2}\},\{y(x)\to -\frac{\sqrt{\frac{36x^6+\frac{362^{2/3}(\cosh \left(3c_1\right)-\sinh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}{x}^2+\cosh \left(6c_1\right)-\sinh \left(6c_1\right)}{x^4}}{x}^2+\sqrt{72x^2+\frac{362^{2/3}(\sinh \left(3c_1\right)-\cosh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}-\frac{9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}{x^2}+\frac{2\cosh \left(6c_1\right)}{x^4}-\frac{2\sinh \left(6c_1\right)}{x^4}-\frac{2(432\cosh \left(3c_1\right)x^6-432\sinh \left(3c_1\right)x^6-\cosh \left(9c_1\right)+\sinh \left(9c_1\right))}{\sqrt{\frac{36x^6+\frac{362^{2/3}(\cosh \left(3c_1\right)-\sinh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}{x}^2+\cosh \left(6c_1\right)-\sinh \left(6c_1\right)}{x^4}}{x}^6}}{x}^2+\cosh \left(3c_1\right)-\sinh \left(3c_1\right)}{18x^2}\},\{y(x)\to \frac{-\sqrt{\frac{36x^6+\frac{362^{2/3}(\cosh \left(3c_1\right)-\sinh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}{x}^2+\cosh \left(6c_1\right)-\sinh \left(6c_1\right)}{x^4}}{x}^2+\sqrt{72x^2+\frac{362^{2/3}(\sinh \left(3c_1\right)-\cosh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}-\frac{9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}{x^2}+\frac{2\cosh \left(6c_1\right)}{x^4}-\frac{2\sinh \left(6c_1\right)}{x^4}-\frac{2(432\cosh \left(3c_1\right)x^6-432\sinh \left(3c_1\right)x^6-\cosh \left(9c_1\right)+\sinh \left(9c_1\right))}{\sqrt{\frac{36x^6+\frac{362^{2/3}(\cosh \left(3c_1\right)-\sinh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}{x}^2+\cosh \left(6c_1\right)-\sinh \left(6c_1\right)}{x^4}}{x}^6}}{x}^2-\cosh \left(3c_1\right)+\sinh \left(3c_1\right)}{18x^2}\},\{y(x)\to \frac{\sqrt{\frac{36x^6+\frac{362^{2/3}(\cosh \left(3c_1\right)-\sinh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}{x}^2+\cosh \left(6c_1\right)-\sinh \left(6c_1\right)}{x^4}}{x}^2-\sqrt{72x^2+\frac{362^{2/3}(\sinh \left(3c_1\right)-\cosh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}-\frac{9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}{x^2}+\frac{2\cosh \left(6c_1\right)}{x^4}-\frac{2\sinh \left(6c_1\right)}{x^4}+\frac{2(432\cosh \left(3c_1\right)x^6-432\sinh \left(3c_1\right)x^6-\cosh \left(9c_1\right)+\sinh \left(9c_1\right))}{\sqrt{\frac{36x^6+\frac{362^{2/3}(\cosh \left(3c_1\right)-\sinh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}{x}^2+\cosh \left(6c_1\right)-\sinh \left(6c_1\right)}{x^4}}{x}^6}}{x}^2-\cosh \left(3c_1\right)+\sinh \left(3c_1\right)}{18x^2}\},\{y(x)\to \frac{\sqrt{\frac{36x^6+\frac{362^{2/3}(\cosh \left(3c_1\right)-\sinh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}{x}^2+\cosh \left(6c_1\right)-\sinh \left(6c_1\right)}{x^4}}{x}^2+\sqrt{72x^2+\frac{362^{2/3}(\sinh \left(3c_1\right)-\cosh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}-\frac{9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}{x^2}+\frac{2\cosh \left(6c_1\right)}{x^4}-\frac{2\sinh \left(6c_1\right)}{x^4}+\frac{2(432\cosh \left(3c_1\right)x^6-432\sinh \left(3c_1\right)x^6-\cosh \left(9c_1\right)+\sinh \left(9c_1\right))}{\sqrt{\frac{36x^6+\frac{362^{2/3}(\cosh \left(3c_1\right)-\sinh \left(3c_1\right))((2x^6-1)\cosh \left(3c_1\right)+(2x^6+1)\sinh \left(3c_1\right))x^4}{\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}}+9\sqrt[3]{2}\sqrt[3]{32x^{12}+40\cosh \left(6c_1\right)x^6-40\sinh \left(6c_1\right)x^6-\cosh \left(12c_1\right)+\sinh \left(12c_1\right)+(\cosh \left(18c_1\right)-\sinh \left(18c_1\right))\sqrt{((16x^6+1)\cosh \left(3c_1\right)+(16x^6-1)\sinh \left(3c_1\right)){}^3(\cosh \left(21c_1\right)+\sinh \left(21c_1\right))}}{x}^2+\cosh \left(6c_1\right)-\sinh \left(6c_1\right)}{x^4}}{x}^6}}{x}^2-\cosh \left(3c_1\right)+\sinh \left(3c_1\right)}{18x^2}\}\}$$

Maple ✓
cpu = 0.029 (sec), leaf count = 37

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}{(3{\mathit{\_}\mathit{T}}^2-1)}^{-\frac{2}{3}},y\left(\mathit{\_}\mathit{T}\right)=-\frac{({\mathit{\_}\mathit{T}}^2-1)\mathit{\_}\mathit{C}\mathit{1}}{2}{(3{\mathit{\_}\mathit{T}}^2-1)}^{-\frac{2}{3}}]\}$$ Mathematica raw input

DSolve[-x + 2*y[x]*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (Cosh[3*C[1]] + Sinh[3*C[1]] - x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] + Sinh[
6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^12 + 40*x
^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1
 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*
C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 
40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*
x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4] - 9*x^2*Sqrt[
(8*x^2)/9 + (2*Cosh[6*C[1]])/(81*x^4) + (2*Sinh[6*C[1]])/(81*x^4) - (4*2^(2/3)*(
2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(9*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[1
2*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] 
+ (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3)) - (2^(1/
3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[1
2*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[1
5*C[1]] + Sinh[15*C[1]])])^(1/3))/(9*x^2) - (2*(-432*x^6*Cosh[3*C[1]] + Cosh[9*C
[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(81*x^6*Sqrt[(36*x^6 + Cosh[6*C[1]]
 + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^1
2 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + 
Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + 
Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*
C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + 
(1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4])])/(1
8*x^2)}, {y[x] -> (Cosh[3*C[1]] + Sinh[3*C[1]] - x^2*Sqrt[(36*x^6 + Cosh[6*C[1]]
 + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(32*x^1
2 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + 
Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + 
Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*
C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + 
(1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^4] + 9*x
^2*Sqrt[(8*x^2)/9 + (2*Cosh[6*C[1]])/(81*x^4) + (2*Sinh[6*C[1]])/(81*x^4) - (4*2
^(2/3)*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(9*(32*x^12 + 40*x^6*Cosh[6*C[1]] 
- Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[
3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3)) 
- (2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] 
- Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3
*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/(9*x^2) - (2*(-432*x^6*Cosh[3*C[1]] + 
Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(81*x^6*Sqrt[(36*x^6 + Cosh
[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))
/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*
C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*
C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - 
Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*
C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^
4])])/(18*x^2)}, {y[x] -> (Cosh[3*C[1]] + Sinh[3*C[1]] + x^2*Sqrt[(36*x^6 + Cosh
[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))
/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*
C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*
C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - 
Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*
C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/x^
4] - 9*x^2*Sqrt[(8*x^2)/9 + (2*Cosh[6*C[1]])/(81*x^4) + (2*Sinh[6*C[1]])/(81*x^4
) - (4*2^(2/3)*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(9*(32*x^12 + 40*x^6*Cosh[
6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^
6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])
^(1/3)) - (2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[
6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*
C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/(9*x^2) + (2*(-432*x^6*Cosh[3*
C[1]] + Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(81*x^6*Sqrt[(36*x^
6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6
*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - 
Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(
Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*
C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)
*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(
1/3))/x^4])])/(18*x^2)}, {y[x] -> (Cosh[3*C[1]] + Sinh[3*C[1]] + x^2*Sqrt[(36*x^
6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] - Sinh[6
*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - 
Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(
Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*
C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)
*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(
1/3))/x^4] + 9*x^2*Sqrt[(8*x^2)/9 + (2*Cosh[6*C[1]])/(81*x^4) + (2*Sinh[6*C[1]])
/(81*x^4) - (4*2^(2/3)*(2*x^6 - Cosh[6*C[1]] - Sinh[6*C[1]]))/(9*(32*x^12 + 40*x
^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1
 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*
C[1]])])^(1/3)) - (2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x
^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)
*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3))/(9*x^2) + (2*(-432*x^6
*Cosh[3*C[1]] + Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(81*x^6*Sqr
t[(36*x^6 + Cosh[6*C[1]] + Sinh[6*C[1]] + (36*2^(2/3)*x^4*(2*x^6 - Cosh[6*C[1]] 
- Sinh[6*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*
C[1]] - Sinh[12*C[1]] + Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[
1]])^3*(Cosh[15*C[1]] + Sinh[15*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6
*Cosh[6*C[1]] - Cosh[12*C[1]] + 40*x^6*Sinh[6*C[1]] - Sinh[12*C[1]] + Sqrt[((1 +
 16*x^6)*Cosh[3*C[1]] + (1 - 16*x^6)*Sinh[3*C[1]])^3*(Cosh[15*C[1]] + Sinh[15*C[
1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> -(Cosh[3*C[1]] - Sinh[3*C[1]] + x^2*Sq
rt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[
3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x
^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18
*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C
[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^
6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*
C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[
1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4] + x^2*Sqrt[72*x^2 + (2*Cosh
[6*C[1]])/x^4 - (2*Sinh[6*C[1]])/x^4 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*C[1]]
)*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh
[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] 
- Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3
*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^12 + 40*x^6*Cosh[6*C
[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Si
nh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Co
sh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^2 - (2*(432*x^6*Cosh[3*C[1]] - Cosh[9*C[
1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^6*Sqrt[(36*x^6 + Cosh[6*C[1]] - S
inh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3
*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1
]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[
((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh
[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]
] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[(
(1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[
21*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> (-Cosh[3*C[1]] + Sinh[3*C[1]] - x
^2*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - 
Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 +
 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Co
sh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sin
h[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 
40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cos
h[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh
[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4] + x^2*Sqrt[72*x^2 + (2
*Cosh[6*C[1]])/x^4 - (2*Sinh[6*C[1]])/x^4 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*
C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6
*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C
[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1
]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^12 + 40*x^6*Cos
h[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]]
 - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^
3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^2 - (2*(432*x^6*Cosh[3*C[1]] - Cosh
[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^6*Sqrt[(36*x^6 + Cosh[6*C[1]
] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*C
osh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[1
2*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*
Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] +
 Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12
*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*S
qrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + 
Sinh[21*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> (-Cosh[3*C[1]] + Sinh[3*C[1]
] + x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1
]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x
^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] 
+ (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6
)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^
12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] +
 (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)
*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4] - x^2*Sqrt[72*x^2
 + (2*Cosh[6*C[1]])/x^4 - (2*Sinh[6*C[1]])/x^4 + (36*2^(2/3)*(-Cosh[3*C[1]] + Si
nh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 4
0*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh
[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[
3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^12 + 40*x^
6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*
C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[
1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^2 + (2*(432*x^6*Cosh[3*C[1]] -
 Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^6*Sqrt[(36*x^6 + Cosh[6
*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x
^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]] - C
osh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[
1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[
1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]] - Co
sh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1
]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1
]] + Sinh[21*C[1]])])^(1/3))/x^4])])/(18*x^2)}, {y[x] -> (-Cosh[3*C[1]] + Sinh[3
*C[1]] + x^2*Sqrt[(36*x^6 + Cosh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[
3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/
(32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C
[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 1
6*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(
32*x^12 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[
1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16
*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4] + x^2*Sqrt[7
2*x^2 + (2*Cosh[6*C[1]])/x^4 - (2*Sinh[6*C[1]])/x^4 + (36*2^(2/3)*(-Cosh[3*C[1]]
 + Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^1
2 + 40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + 
(Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*
Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^12 + 
40*x^6*Cosh[6*C[1]] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cos
h[18*C[1]] - Sinh[18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh
[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^2 + (2*(432*x^6*Cosh[3*C[
1]] - Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^6*Sqrt[(36*x^6 + C
osh[6*C[1]] - Sinh[6*C[1]] + (36*2^(2/3)*x^4*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 
+ 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^12 + 40*x^6*Cosh[6*C[1]
] - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[
18*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[
21*C[1]] + Sinh[21*C[1]])])^(1/3) + 9*2^(1/3)*x^2*(32*x^12 + 40*x^6*Cosh[6*C[1]]
 - Cosh[12*C[1]] - 40*x^6*Sinh[6*C[1]] + Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[1
8*C[1]])*Sqrt[((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[2
1*C[1]] + Sinh[21*C[1]])])^(1/3))/x^4])])/(18*x^2)}}

Maple raw input

dsolve(x*diff(y(x),x)^2+2*y(x)*diff(y(x),x)-x = 0, y(x),'implicit')

Maple raw output

[x(_T) = 1/(3*_T^2-1)^(2/3)*_T*_C1, y(_T) = -1/2*(_T^2-1)*_C1/(3*_T^2-1)^(2/3)]