54.1.257 problem 262

Internal problem ID [11571]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 262
Date solved : Tuesday, September 30, 2025 at 09:20:53 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

\begin{align*} \left (2 x^{2} y-x^{3}\right ) y^{\prime }+y^{3}-4 x y^{2}+2 x^{3}&=0 \end{align*}

✓ Maple. Time used: 0.124 (sec). Leaf size: 65

ode:=(2*x^2*y(x)-x^3)*diff(y(x),x)+y(x)^3-4*x*y(x)^2+2*x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x \left (2 c_1 \,x^{2}-\sqrt {3 c_1 \,x^{2}+1}\right )}{c_1 \,x^{2}-1} \\ y &= \frac {x \left (2 c_1 \,x^{2}+\sqrt {3 c_1 \,x^{2}+1}\right )}{c_1 \,x^{2}-1} \\ \end{align*}

✓ Mathematica. Time used: 13.206 (sec). Leaf size: 132

ode=(2*x^2*y[x]-x^3)*D[y[x],x]+y[x]^3-4*x*y[x]^2+2*x^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2 x^3-\sqrt {e^{2 c_1} x^2 \left (-3 x^2+e^{2 c_1}\right )}}{x^2+e^{2 c_1}}\\ y(x)&\to \frac {2 x^3+\sqrt {e^{2 c_1} x^2 \left (-3 x^2+e^{2 c_1}\right )}}{x^2+e^{2 c_1}}\\ y(x)&\to 2 x\\ y(x)&\to -\sqrt {x^2}\\ y(x)&\to \sqrt {x^2} \end{align*}

✓ Sympy. Time used: 2.164 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**3 - 4*x*y(x)**2 + (-x**3 + 2*x**2*y(x))*Derivative(y(x), x) + y(x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {x \left (2 x^{2} - \sqrt {C_{1} \left (C_{1} + 3 x^{2}\right )}\right )}{- C_{1} + x^{2}}, \ y{\left (x \right )} = \frac {x \left (2 x^{2} + \sqrt {C_{1} \left (C_{1} + 3 x^{2}\right )}\right )}{- C_{1} + x^{2}}\right ] \]