63.1.29 problem 46

Internal problem ID [15469]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 46
Date solved : Thursday, October 02, 2025 at 10:15:40 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 58

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

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

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 63

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

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

✓ Sympy. Time used: 0.953 (sec). Leaf size: 68

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

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