Internal problem ID [14303]
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: 15.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
\[ \boxed {y^{3}+\left (3 t y^{2}+4\right ) y^{\prime }=-2 t} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 394
dsolve((2*t+y(t)^3)+(3*t*y(t)^2+4)*diff(y(t),t)=0,y(t), singsol=all)
\begin{align*} y \left (t \right ) &= -\frac {2 \,12^{\frac {1}{3}} \left (12^{\frac {1}{3}} t -\frac {\left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {2}{3}}}{4}\right )}{3 \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {1}{3}} t} \\ y \left (t \right ) &= -\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (4 i 2^{\frac {2}{3}} 3^{\frac {5}{6}} t +i \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {2}{3}} \sqrt {3}-4 \,3^{\frac {1}{3}} 2^{\frac {2}{3}} t +\left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {2}{3}}\right )}{12 t \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {1}{3}}} \\ y \left (t \right ) &= \frac {\left (4 \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} t +\left (i \sqrt {3}-1\right ) \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{12 \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {1}{3}} t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 23.63 (sec). Leaf size: 369
DSolve[(2*t+y[t]^3)+(3*t*y[t]^2+4)*y'[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {\sqrt [3]{-27 t^4+27 c_1 t^2+\sqrt {6912 t^3+729 \left (t^4-c_1 t^2\right ){}^2}}}{3 \sqrt [3]{2} t}-\frac {4 \sqrt [3]{2}}{\sqrt [3]{-27 t^4+27 c_1 t^2+\sqrt {6912 t^3+729 \left (t^4-c_1 t^2\right ){}^2}}} \\ y(t)\to \frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{-27 t^4+27 c_1 t^2+\sqrt {6912 t^3+729 \left (t^4-c_1 t^2\right ){}^2}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-27 t^4+27 c_1 t^2+\sqrt {6912 t^3+729 \left (t^4-c_1 t^2\right ){}^2}}}{6 \sqrt [3]{2} t} \\ y(t)\to \frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right )}{\sqrt [3]{-27 t^4+27 c_1 t^2+\sqrt {6912 t^3+729 \left (t^4-c_1 t^2\right ){}^2}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 t^4+27 c_1 t^2+\sqrt {6912 t^3+729 \left (t^4-c_1 t^2\right ){}^2}}}{6 \sqrt [3]{2} t} \\ \end{align*}