23.2.158 problem 161

Internal problem ID [5513]
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 : 161
Date solved : Tuesday, September 30, 2025 at 12:46:05 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.402 (sec). Leaf size: 123

ode:=x^2*diff(y(x),x)^2-4*x*(2+y(x))*diff(y(x),x)+4*(2+y(x))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -2 \\ y &= \frac {x \left (2 \sqrt {2}\, c_1 +x \right )}{c_1^{2}} \\ y &= \frac {\left (-2 \sqrt {2}\, c_1 +x \right ) x}{c_1^{2}} \\ y &= \frac {\left (-8 c_1^{2}+x^{2}\right ) x \left (-2 \sqrt {2}\, c_1 +x \right )}{\left (-4 \sqrt {2}\, c_1 x +8 c_1^{2}+x^{2}\right ) c_1^{2}} \\ y &= \frac {\left (-8 c_1^{2}+x^{2}\right ) \left (2 \sqrt {2}\, c_1 +x \right ) x}{\left (4 \sqrt {2}\, c_1 x +8 c_1^{2}+x^{2}\right ) c_1^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.125 (sec). Leaf size: 69

ode=x^2 (D[y[x],x])^2-4 x(2+y[x])D[y[x],x]+4(2+y[x])y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-c_1} x \left (x-2 \sqrt {2} e^{\frac {c_1}{2}}\right )\\ y(x)&\to e^{c_1} x^2-2 \sqrt {2} e^{\frac {c_1}{2}} x\\ y(x)&\to -2\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 20.592 (sec). Leaf size: 160

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x)**2 - 4*x*(y(x) + 2)*Derivative(y(x), x) + (4*y(x) + 8)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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