32.4.23 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.15, page 90

Internal problem ID [5834]
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 10.15, page 90
Date solved : Sunday, March 30, 2025 at 10:19:05 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{2}+12 x^{2} y+\left (2 x y+4 x^{3}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 51

ode:=y(x)^2+12*x^2*y(x)+(2*x*y(x)+4*x^3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {-2 x^{3}+\sqrt {4 x^{6}+c_1 x}}{x} \\ y &= \frac {-2 x^{3}-\sqrt {4 x^{6}+c_1 x}}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.513 (sec). Leaf size: 58

ode=(y[x]^2+12*x^2*y[x])+(2*x*y[x]+4*x^3)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {2 x^3+\sqrt {x \left (4 x^5+c_1\right )}}{x} \\ y(x)\to \frac {-2 x^3+\sqrt {x \left (4 x^5+c_1\right )}}{x} \\ \end{align*}

✓ Sympy. Time used: 9.423 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(12*x**2*y(x) + (4*x**3 + 2*x*y(x))*Derivative(y(x), x) + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x^{2} \left (- \sqrt {\frac {C_{1}}{x^{5}} + 4} - 2\right ), \ y{\left (x \right )} = x^{2} \left (\sqrt {\frac {C_{1}}{x^{5}} + 4} - 2\right )\right ] \]