49.22.9 problem 2(a)

Internal problem ID [7755]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 5. Existence and uniqueness of solutions to first order equations. Page 198
Problem number : 2(a)
Date solved : Monday, January 27, 2025 at 03:12:05 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 71

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

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

✓ Solution by Mathematica

Time used: 0.301 (sec). Leaf size: 215

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

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