Internal problem ID [3693]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 968.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (x -2 y\right ) \left (y^{\prime }\right )^{2}-2 y^{\prime } y x -2 y x +y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 109
dsolve(x*(x-2*y(x))*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-2*x*y(x)+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \RootOf \left (-2 \ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {-2 \textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{3}-4 \textit {\_a}^{2}+2 \textit {\_a}}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-2 \ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{3}-4 \textit {\_a}^{2}+2 \textit {\_a}}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} \right )+2 c_{1}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.524 (sec). Leaf size: 167
DSolve[x(x-2 y[x]) (y'[x])^2-2 x y[x] y'[x]-2 x y[x]+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x \left (x+2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}} \\ y(x)\to \sqrt {-x \left (x+2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}} \\ y(x)\to e^{\frac {c_1}{2}}-\sqrt {x \left (-x+2 e^{\frac {c_1}{2}}\right )} \\ y(x)\to \sqrt {x \left (-x+2 e^{\frac {c_1}{2}}\right )}+e^{\frac {c_1}{2}} \\ y(x)\to -\sqrt {-x^2} \\ y(x)\to \sqrt {-x^2} \\ \end{align*}