78.5.11 problem 2 (k)

Internal problem ID [18083]
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 (k)
Date solved : Monday, March 31, 2025 at 05:07:27 PM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 79

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

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

✓ Mathematica. Time used: 4.411 (sec). Leaf size: 167

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

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

✓ Sympy. Time used: 2.445 (sec). Leaf size: 78

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

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