23.2.161 problem 164

Internal problem ID [5516]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 164
Date solved : Tuesday, September 30, 2025 at 12:47:33 PM
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*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 101

ode:=x^2*diff(y(x),x)^2+(y(x)+2*x)*y(x)*diff(y(x),x)+y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -4 x \\ y &= 0 \\ y &= \frac {c_1^{2}}{2 x -\sqrt {2}\, c_1} \\ y &= \frac {c_1^{2}}{2 x +\sqrt {2}\, c_1} \\ y &= \frac {2 c_1^{2} \left (-\sqrt {2}\, c_1 +x \right )}{-2 c_1^{2}+x^{2}} \\ y &= \frac {2 c_1^{2} \left (\sqrt {2}\, c_1 +x \right )}{-2 c_1^{2}+x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 3.633 (sec). Leaf size: 117

ode=x^2 (D[y[x],x])^2+(2 x+y[x])y[x] D[y[x],x]+y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 6.801 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x)**2 + (2*x + y(x))*y(x)*Derivative(y(x), x) + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {e^{2 C_{1}}}{2 \left (2 x + e^{C_{1}}\right )} \]