74.6.17 problem 18

Internal problem ID [16040]
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
Date solved : Tuesday, January 28, 2025 at 08:28:43 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

\begin{align*} 2 t y^{3}+\left (1+3 t^{2} y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 249

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 &= -\frac {12^{{1}/{3}} \left (c_{1}^{2} 12^{{1}/{3}}-{\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{{2}/{3}}\right )}{6 c_{1} t {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{{1}/{3}}} \\ y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{{2}/{3}}+\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}} c_{1}^{2}\right )}{12 {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{{1}/{3}} t c_{1}} \\ y &= \frac {2^{{2}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) c_{1}^{2}\right ) 3^{{1}/{3}}}{12 {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{{1}/{3}} t c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 37.052 (sec). Leaf size: 364

DSolve[2*t*y[t]^3+(1+3*t^2*y[t]^2)*D[y[t],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*}