29.21.3 problem 579

Internal problem ID [5173]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 21
Problem number : 579
Date solved : Sunday, March 30, 2025 at 06:47:00 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.180 (sec). Leaf size: 65

ode:=x^2*(x-2*y(x))*diff(y(x),x) = 2*x^3-4*x*y(x)^2+y(x)^3; 
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: 14.095 (sec). Leaf size: 132

ode=x^2(x-2 y[x])D[y[x],x]==2 x^3-4 x y[x]^2+y[x]^3; 
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: 3.363 (sec). Leaf size: 51

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

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