29.31.3 problem 902

Internal problem ID [5482]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 902
Date solved : Sunday, March 30, 2025 at 08:17:00 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.240 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.211 (sec). Leaf size: 64

ode=x^2 (D[y[x],x])^2-x(x-2 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 {e^{-4 c_1}-2 i e^{-2 c_1} x}{4 x} \\ y(x)\to \frac {2 i e^{-2 c_1} x+e^{-4 c_1}}{4 x} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 49.460 (sec). Leaf size: 160

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

\[ \left [ y{\left (x \right )} = \frac {C_{1}}{2 x} - \frac {\sqrt {2} \sqrt {- C_{1}}}{2}, \ y{\left (x \right )} = \frac {C_{1}}{2 x} + \frac {\sqrt {2} \sqrt {- C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1}}}{2} - \frac {C_{1}}{2 x}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1}}}{2} - \frac {C_{1}}{2 x}, \ y{\left (x \right )} = \frac {C_{1}}{2 x} - \frac {\sqrt {2} \sqrt {- C_{1}}}{2}, \ y{\left (x \right )} = \frac {C_{1}}{2 x} + \frac {\sqrt {2} \sqrt {- C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1}}}{2} - \frac {C_{1}}{2 x}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1}}}{2} - \frac {C_{1}}{2 x}\right ] \]