30.4.2 problem 2

Internal problem ID [7485]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.5, Special Integrating Factors. Exercises. page 69
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 04:39:20 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.074 (sec). Leaf size: 1038

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 60.13 (sec). Leaf size: 325

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

\begin{align*} y(x)&\to 0\\ y(x)&\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{\sqrt {-4+\left (2-\frac {27 e^{c_1}}{x^2}\right ){}^2}+\frac {27 e^{c_1}}{x^2}-2}+\frac {2 \sqrt [3]{2}}{\sqrt [3]{\sqrt {-4+\left (2-\frac {27 e^{c_1}}{x^2}\right ){}^2}+\frac {27 e^{c_1}}{x^2}-2}}-2\right )\\ y(x)&\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {-4+\left (2-\frac {27 e^{c_1}}{x^2}\right ){}^2}+\frac {27 e^{c_1}}{x^2}-2}-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{\sqrt {-4+\left (2-\frac {27 e^{c_1}}{x^2}\right ){}^2}+\frac {27 e^{c_1}}{x^2}-2}}-4\right )\\ y(x)&\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-4+\left (2-\frac {27 e^{c_1}}{x^2}\right ){}^2}+\frac {27 e^{c_1}}{x^2}-2}+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right )}{\sqrt [3]{\sqrt {-4+\left (2-\frac {27 e^{c_1}}{x^2}\right ){}^2}+\frac {27 e^{c_1}}{x^2}-2}}-4\right ) \end{align*}

✗ Sympy

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

Timed Out