32.4.24 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.16, page 90

Internal problem ID [5835]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number : Recognizable Exact Differential equations. Integrating factors. Exercise 10.16, page 90
Date solved : Sunday, March 30, 2025 at 10:19:07 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 63

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

\begin{align*} y &= \frac {-4 c_1 \,x^{2}-\sqrt {-2 x^{4} c_1^{2}+6}}{6 c_1 x} \\ y &= \frac {-4 c_1 \,x^{2}+\sqrt {-2 x^{4} c_1^{2}+6}}{6 c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 1.847 (sec). Leaf size: 135

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

\begin{align*} y(x)\to -\frac {4 x^2+\sqrt {-2 x^4+6 e^{4 c_1}}}{6 x} \\ y(x)\to \frac {-4 x^2+\sqrt {-2 x^4+6 e^{4 c_1}}}{6 x} \\ y(x)\to -\frac {\sqrt {2} \sqrt {-x^4}+4 x^2}{6 x} \\ y(x)\to \frac {\sqrt {2} \sqrt {-x^4}-4 x^2}{6 x} \\ \end{align*}

✓ Sympy. Time used: 1.659 (sec). Leaf size: 42

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

\[ \left [ y{\left (x \right )} = - \frac {2 x}{3} - \frac {\sqrt {C_{1} - 2 x^{4}}}{6 x}, \ y{\left (x \right )} = - \frac {2 x}{3} + \frac {\sqrt {C_{1} - 2 x^{4}}}{6 x}\right ] \]