78.5.11 problem 2 (k)

Internal problem ID [18154]
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 : Tuesday, January 28, 2025 at 11:34:19 AM
CAS classification : [_rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 79

dsolve(( x^3+x*y(x)^3 ) +( 3*y(x)^2   )*diff(y(x),x)=0,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 ({\mathrm e}^{-\frac {x^{2}}{2}} c_1 -x^{2}+2\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left ({\mathrm e}^{-\frac {x^{2}}{2}} c_1 -x^{2}+2\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

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

DSolve[( x^2+x*y[x]^3 )+( 3*y[x]^2 )*D[y[x],x]==0,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*}