75.6.25 problem 158

Internal problem ID [16702]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 6. Linear equations of the first order. The Bernoulli equation. Exercises page 54
Problem number : 158
Date solved : Thursday, March 13, 2025 at 08:32:30 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} 3 x y^{2} y^{\prime }-2 y^{3}&=x^{3} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 55

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

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

✓ Mathematica. Time used: 0.204 (sec). Leaf size: 66

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

\begin{align*} y(x)\to x^{2/3} \sqrt [3]{x+c_1} \\ y(x)\to -\sqrt [3]{-1} x^{2/3} \sqrt [3]{x+c_1} \\ y(x)\to (-1)^{2/3} x^{2/3} \sqrt [3]{x+c_1} \\ \end{align*}

✓ Sympy. Time used: 1.245 (sec). Leaf size: 58

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

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