54.1.430 problem 441

Internal problem ID [11744]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 441
Date solved : Tuesday, September 30, 2025 at 10:15:12 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.802 (sec). Leaf size: 69

ode:=x^2*diff(y(x),x)^2-4*x*(y(x)+2)*diff(y(x),x)+4*y(x)*(y(x)+2) = 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 {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}} \\ \end{align*}

✓ Mathematica. Time used: 0.111 (sec). Leaf size: 69

ode=4*y[x]*(2 + y[x]) - 4*x*(2 + y[x])*D[y[x],x] + x^2*D[y[x],x]^2==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: 21.063 (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)
 

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