18.1.9 problem Problem 14.5 (c)

Internal problem ID [3465]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 14, First order ordinary differential equations. 14.4 Exercises, page 490
Problem number : Problem 14.5 (c)
Date solved : Monday, January 27, 2025 at 07:37:51 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} \left (x +y^{3}\right ) y^{\prime }&=y \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 220

dsolve((x+y(x)^3)*diff(y(x),x)=y(x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}-6 c_{1}}{3 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}+6 i c_{1} \sqrt {3}+\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}-6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}+6 i c_{1} \sqrt {3}-\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}+6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.697 (sec). Leaf size: 263

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

\begin{align*} y(x)\to \frac {2\ 3^{2/3} c_1-\sqrt [3]{3} \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}}{3 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\ y(x)\to \frac {\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}-2 \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) c_1}{6 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\ y(x)\to \frac {\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}-2 \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) c_1}{6 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\ y(x)\to 0 \\ \end{align*}