Internal problem ID [14306]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number: 18.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _exact, _rational]
\[ \boxed {2 y^{3} t +\left (1+3 t^{2} y^{2}\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 252
dsolve(2*t*y(t)^3+(1+3*t^2*y(t)^2)*diff(y(t),t)=0,y(t), singsol=all)
\begin{align*} y \left (t \right ) &= \frac {12^{\frac {1}{3}} \left (-c_{1}^{2} 12^{\frac {1}{3}}+{\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {2}{3}}\right )}{6 c_{1} t {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {1}{3}}} \\ y \left (t \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) c_{1}^{2} 2^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{12 {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1} t} \\ y \left (t \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) c_{1}^{2} 2^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{12 {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1} t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 28.553 (sec). Leaf size: 364
DSolve[2*t*y[t]^3+(1+3*t^2*y[t]^2)*y'[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {-2 \sqrt [3]{3}+\frac {\sqrt [3]{2} \left (9 c_1 t^4+\sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}\right ){}^{2/3}}{t^2}}{6^{2/3} \sqrt [3]{9 c_1 t^4+\sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}}} \\ y(t)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) t^2+i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (18 c_1 t^4+2 \sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}\right ){}^{2/3}}{12 t^2 \sqrt [3]{9 c_1 t^4+\sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}}} \\ y(t)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) t^2+\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (18 c_1 t^4+2 \sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}\right ){}^{2/3}}{12 t^2 \sqrt [3]{9 c_1 t^4+\sqrt {3} \sqrt {t^6 \left (4+27 c_1{}^2 t^2\right )}}} \\ y(t)\to 0 \\ \end{align*}