29.31.13 problem 912

Internal problem ID [5492]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 912
Date solved : Monday, January 27, 2025 at 11:31:30 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.074 (sec). Leaf size: 124

dsolve(x^2*diff(y(x),x)^2+(2*x-y(x))*y(x)*diff(y(x),x)+y(x)^2 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= 4 x \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {2 c_{1}^{2} \left (-\sqrt {2}\, c_{1} +x \right )}{-2 c_{1}^{2}+x^{2}} \\ y \left (x \right ) &= -\frac {2 c_{1}^{2} \left (\sqrt {2}\, c_{1} +x \right )}{-2 c_{1}^{2}+x^{2}} \\ y \left (x \right ) &= \frac {\sqrt {2}\, c_{1}^{3}-2 c_{1}^{2} x}{-2 c_{1}^{2}+4 x^{2}} \\ y \left (x \right ) &= \frac {c_{1}^{2} \left (\sqrt {2}\, c_{1} +2 x \right )}{2 c_{1}^{2}-4 x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.008 (sec). Leaf size: 61

DSolve[x^2 (D[y[x],x])^2+(2 x-y[x])y[x] D[y[x],x]+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {4 e^{-2 c_1}}{-2+e^{2 c_1} x} \\ y(x)\to -\frac {4 e^{-2 c_1}}{2+e^{2 c_1} x} \\ y(x)\to 0 \\ y(x)\to 4 x \\ \end{align*}