66.1.49 problem Problem 63

Internal problem ID [13825]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 63
Date solved : Monday, March 31, 2025 at 08:14:22 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational, _Bernoulli]

\begin{align*} 3 x y^{2} y^{\prime }+y^{3}-2 x&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 71

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

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

✓ Mathematica. Time used: 0.199 (sec). Leaf size: 72

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

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

✓ Sympy. Time used: 1.205 (sec). Leaf size: 53

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

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