Internal problem ID [7960]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 380.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+2 x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 690
dsolve(diff(y(x),x)^2+2*x*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}\right )^{2}+2 \left (\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}\right ) x \\ y \relax (x ) = \left (-\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 \left (-\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ y \relax (x ) = \left (-\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 \left (-\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.183 (sec). Leaf size: 1120
DSolve[-y[x] + 2*x*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^4+\left (x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}+x^2 \sqrt [3]{x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}+8 e^{3 c_1} x}{4 \sqrt [3]{x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ y(x)\to \frac {1}{72} \left (-18 x^2+\frac {\left (9+9 i \sqrt {3}\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}}+\left (9-9 i \sqrt {3}\right ) \sqrt [3]{x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{72} \left (-18 x^2+\frac {\left (9-9 i \sqrt {3}\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}}+\left (9+9 i \sqrt {3}\right ) \sqrt [3]{x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}\right ) \\ y(x)\to -\frac {x^4+\left (x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}+x^2 \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}-8 e^{3 c_1} x}{4 \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) x^4+\left (1-i \sqrt {3}\right ) \left (x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}-2 x^2 \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}+\left (-8-8 i \sqrt {3}\right ) e^{3 c_1} x}{8 \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^4+\left (1+i \sqrt {3}\right ) \left (x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}\right ){}^{2/3}-2 x^2 \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}+8 i \left (\sqrt {3}+i\right ) e^{3 c_1} x}{8 \sqrt [3]{x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}-8 e^{6 c_1}}} \\ \end{align*}