60.1.480 problem 483

Internal problem ID [10494]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 483
Date solved : Monday, January 27, 2025 at 08:15:32 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.120 (sec). Leaf size: 103

dsolve((2*x*y(x)-x^2)*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 &= 0 \\ y &= \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 &= \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: 3.982 (sec). Leaf size: 167

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