49.22.1 problem 1(a)

Internal problem ID [7747]
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 : 1(a)
Date solved : Monday, January 27, 2025 at 03:11:59 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 185

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

\begin{align*} y &= \frac {-12 c_{1} x^{2}+\left (108+12 \sqrt {12 c_{1}^{3} x^{6}+81}\right )^{{2}/{3}}}{6 \left (108+12 \sqrt {12 c_{1}^{3} x^{6}+81}\right )^{{1}/{3}} \sqrt {c_{1}}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (108+12 \sqrt {12 c_{1}^{3} x^{6}+81}\right )^{{1}/{3}}}{12 \sqrt {c_{1}}}-\frac {\sqrt {c_{1}}\, x^{2} \left (i \sqrt {3}-1\right )}{\left (108+12 \sqrt {12 c_{1}^{3} x^{6}+81}\right )^{{1}/{3}}} \\ y &= \frac {\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {12 c_{1}^{3} x^{6}+81}\right )^{{1}/{3}}}{12 \sqrt {c_{1}}}+\frac {\sqrt {c_{1}}\, x^{2} \left (1+i \sqrt {3}\right )}{\left (108+12 \sqrt {12 c_{1}^{3} x^{6}+81}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 23.825 (sec). Leaf size: 441

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

\begin{align*} y(x)\to \frac {-2 \sqrt [3]{3} x^2+\sqrt [3]{2} \left (\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}}} \\ y(x)\to \frac {i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^2}{12 \sqrt [3]{\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}}} \\ y(x)\to \frac {2^{2/3} \sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^2}{12 \sqrt [3]{\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}}} \\ y(x)\to 0 \\ y(x)\to \frac {\sqrt [3]{x^6}-x^2}{\sqrt {3} \sqrt [6]{x^6}} \\ y(x)\to \frac {\left (\sqrt {3}+3 i\right ) x^2-\left (\sqrt {3}-3 i\right ) \sqrt [3]{x^6}}{6 \sqrt [6]{x^6}} \\ y(x)\to \frac {1}{6} \sqrt [6]{x^6} \left (\frac {\left (\sqrt {3}-3 i\right ) \left (x^6\right )^{2/3}}{x^4}-\sqrt {3}-3 i\right ) \\ \end{align*}