29.31.12 problem 911

Internal problem ID [5491]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 911
Date solved : Monday, January 27, 2025 at 11:30:15 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.073 (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 {c_{1}^{2} \left (\sqrt {2}\, c_{1} -2 x \right )}{2 c_{1}^{2}-4 x^{2}} \\ y \left (x \right ) &= \frac {\sqrt {2}\, c_{1}^{3}+2 c_{1}^{2} x}{-2 c_{1}^{2}+4 x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.686 (sec). Leaf size: 117

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 x \left (1+\tanh \left (-\frac {\log (x)}{2}+c_1\right )\right ){}^2}{3 \tanh ^2\left (-\frac {\log (x)}{2}+c_1\right )-2 \tanh \left (-\frac {\log (x)}{2}+c_1\right )-1} \\ y(x)\to \frac {4 x \left (-1+\tanh \left (\frac {1}{2} (\log (x)-2 c_1)\right )\right ){}^2}{3 \tanh ^2\left (-\frac {\log (x)}{2}+c_1\right )-2 \tanh \left (-\frac {\log (x)}{2}+c_1\right )-1} \\ y(x)\to 0 \\ y(x)\to -4 x \\ \end{align*}