43.22.9 problem 2(a)

Internal problem ID [9041]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 5. Existence and uniqueness of solutions to first order equations. Page 198
Problem number : 2(a)
Date solved : Tuesday, September 30, 2025 at 06:02:21 PM
CAS classification : [_separable]

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

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

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

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

✓ Mathematica. Time used: 0.185 (sec). Leaf size: 215

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

\begin{align*} y(x)&\to -\frac {\sqrt [3]{-\frac {1}{3}} \sqrt [3]{-2 x^3+e^{9 c_1}}}{x}\\ y(x)&\to \frac {\sqrt [3]{-2 x^3+e^{9 c_1}}}{\sqrt [3]{3} x}\\ y(x)&\to \frac {(-1)^{2/3} \sqrt [3]{-2 x^3+e^{9 c_1}}}{\sqrt [3]{3} x}\\ y(x)&\to \sqrt [3]{-\frac {2}{3}}\\ y(x)&\to -\sqrt [3]{\frac {2}{3}}\\ y(x)&\to -(-1)^{2/3} \sqrt [3]{\frac {2}{3}}\\ y(x)&\to \frac {\sqrt [3]{-\frac {2}{3}} x^2}{\left (-x^3\right )^{2/3}}\\ y(x)&\to \frac {\sqrt [3]{\frac {2}{3}} \sqrt [3]{-x^3}}{x}\\ y(x)&\to \frac {(-1)^{2/3} \sqrt [3]{\frac {2}{3}} \sqrt [3]{-x^3}}{x} \end{align*}

✓ Sympy. Time used: 1.196 (sec). Leaf size: 78

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

\[ \left [ y{\left (x \right )} = \frac {\left (- \sqrt [3]{-1} - \left (-1\right )^{\frac {5}{6}} \sqrt {3}\right ) \sqrt [3]{\frac {C_{1}}{x^{2}} + 1}}{2}, \ y{\left (x \right )} = \frac {\left (- \sqrt [3]{-1} + \left (-1\right )^{\frac {5}{6}} \sqrt {3}\right ) \sqrt [3]{\frac {C_{1}}{x^{2}} + 1}}{2}, \ y{\left (x \right )} = \sqrt [3]{\frac {C_{1}}{x^{2}} - 1}\right ] \]