69.12.17 problem 291

Internal problem ID [18170]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 291
Date solved : Thursday, October 02, 2025 at 03:06:46 PM
CAS classification : [[_homogeneous, `class D`], _rational, _Bernoulli]

\begin{align*} x y^{2} y^{\prime }-y^{3}&=\frac {x^{4}}{3} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 49

ode:=x*y(x)^2*diff(y(x),x)-y(x)^3 = 1/3*x^4; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 54

ode=x*y[x]^2*D[y[x],x]-y[x]^3==1/3*x^4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.838 (sec). Leaf size: 58

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

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