Internal problem ID [5326]
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).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {2 x y+\left (x^{2}+3 y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.056 (sec). Leaf size: 257
dsolve(2*x*y(x)+(x^2+3*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2} c_{1}}{\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{12}+\frac {x^{2} c_{1}}{\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2} c_{1}}{\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ y \relax (x ) = \frac {-\frac {\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{12}+\frac {x^{2} c_{1}}{\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2} c_{1}}{\left (108+12 \sqrt {12 x^{6} c_{1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.603 (sec). Leaf size: 396
DSolve[2*x*y[x]+(x^2+3*y[x]^2)*y'[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 {\sqrt [3]{-1} \left (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}\right )}{6^{2/3} \sqrt [3]{\sqrt {12 x^6+81 e^{2 c_1}}+9 e^{c_1}}} \\ y(x)\to -\frac {\sqrt [3]{-1} \left (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}\right )}{6^{2/3} \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 {\left (\sqrt {3}+3 i\right ) x^2-\left (\sqrt {3}-3 i\right ) \sqrt [3]{x^6}}{6 \sqrt [6]{x^6}} \\ \end{align*}