50.6.11 problem 1(k)

Internal problem ID [7903]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.8. Integrating Factors. Page 32
Problem number : 1(k)
Date solved : Monday, January 27, 2025 at 03:31:40 PM
CAS classification : [_rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.007 (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: 11.728 (sec). Leaf size: 95

DSolve[(x^3+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]{-x^2+c_1 e^{-\frac {x^2}{2}}+2} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{-x^2+c_1 e^{-\frac {x^2}{2}}+2} \\ y(x)\to (-1)^{2/3} \sqrt [3]{-x^2+c_1 e^{-\frac {x^2}{2}}+2} \\ \end{align*}