29.33.6 problem 968

Internal problem ID [5545]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 968
Date solved : Monday, January 27, 2025 at 11:43:29 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x \left (x -2 y\right ) {y^{\prime }}^{2}-2 x y y^{\prime }-2 y x +y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.175 (sec). Leaf size: 103

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 \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}-2 \textit {\_a}^{2}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.088 (sec). Leaf size: 167

DSolve[x(x-2 y[x]) (D[y[x],x])^2-2 x y[x] D[y[x],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*}