Internal problem ID [10197]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
25. Equations solvable for \(y\). Page 52
Problem number: Ex 6.
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 y^{\prime } x -y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (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 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}\right )^{2}+2 x \left (\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}\right ) \\ y \relax (x ) = \left (-\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+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 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 x \left (-\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+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 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) \\ y \relax (x ) = \left (-\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+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 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 x \left (-\frac {\left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+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 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 c_{1}-x^{3}+2 \sqrt {-3 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.099 (sec). Leaf size: 927
DSolve[(y'[x])^2+2*x*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (-x^2+\frac {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}}}+\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}}}+9 i \left (\sqrt {3}+i\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 {9 i \left (\sqrt {3}+i\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}}}-9 \left (1+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}{4} \left (-x^2+\frac {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}}}+\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}}}+9 i \left (\sqrt {3}+i\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 {9 i \left (\sqrt {3}+i\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}}}-9 \left (1+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 ) \\ \end{align*}