72.5.3 problem 2 (c)

Internal problem ID [19432]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 9 (Integrating Factors). Problems at page 80
Problem number : 2 (c)
Date solved : Thursday, October 02, 2025 at 04:26:49 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x y^{\prime }+y+3 x^{3} y^{4} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 133

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

\begin{align*} y &= -\frac {\sqrt {6}\, \sqrt {\left (x -\sqrt {12 c_1^{2}+x^{2}}\right ) c_1 x}}{6 c_1 x} \\ y &= \frac {\sqrt {6}\, \sqrt {\left (x -\sqrt {12 c_1^{2}+x^{2}}\right ) c_1 x}}{6 c_1 x} \\ y &= -\frac {\sqrt {6}\, \sqrt {c_1 x \left (x +\sqrt {12 c_1^{2}+x^{2}}\right )}}{6 c_1 x} \\ y &= \frac {\sqrt {6}\, \sqrt {c_1 x \left (x +\sqrt {12 c_1^{2}+x^{2}}\right )}}{6 c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 8.936 (sec). Leaf size: 166

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

\begin{align*} y(x)&\to -\frac {\sqrt {c_1-\frac {\sqrt {x^2 \left (3+c_1{}^2 x^2\right )}}{x^2}}}{\sqrt {3}}\\ y(x)&\to \frac {\sqrt {c_1-\frac {\sqrt {x^2 \left (3+c_1{}^2 x^2\right )}}{x^2}}}{\sqrt {3}}\\ y(x)&\to -\frac {\sqrt {\frac {\sqrt {x^2 \left (3+c_1{}^2 x^2\right )}}{x^2}+c_1}}{\sqrt {3}}\\ y(x)&\to \frac {\sqrt {\frac {\sqrt {x^2 \left (3+c_1{}^2 x^2\right )}}{x^2}+c_1}}{\sqrt {3}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 5.047 (sec). Leaf size: 126

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {e^{C_{1}} - \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {e^{C_{1}} - \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {e^{C_{1}} + \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {e^{C_{1}} + \frac {\sqrt {x^{2} e^{2 C_{1}} + 12}}{x}}}{6}\right ] \]