71.7.12 problem 16

Internal problem ID [14433]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.3.3, page 71
Problem number : 16
Date solved : Tuesday, January 28, 2025 at 06:30:56 AM
CAS classification : [_separable]

\begin{align*} x \left (1-y^{3}\right )-3 y^{2} y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 64

dsolve(x*(1-y(x)^3)-3*y(x)^2*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.923 (sec). Leaf size: 111

DSolve[x*(1-y[x]^3)-3*y[x]^2*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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