74.6.14 problem 15
Internal
problem
ID
[16037]
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
Date
solved
:
Tuesday, January 28, 2025 at 08:28:34 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} 2 t +y^{3}+\left (3 t y^{2}+4\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 390
dsolve((2*t+y(t)^3)+(3*t*y(t)^2+4)*diff(y(t),t)=0,y(t), singsol=all)
\begin{align*}
y &= -\frac {2 \,12^{{1}/{3}} \left (12^{{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 )^{{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 )^{{1}/{3}} t} \\
y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (4 i 2^{{2}/{3}} 3^{{5}/{6}} t +i \sqrt {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 )^{{2}/{3}}-4 \,2^{{2}/{3}} 3^{{1}/{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 )^{{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 )^{{1}/{3}}} \\
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (4 \,2^{{2}/{3}} \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) 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 )^{{2}/{3}}\right )}{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 )^{{1}/{3}} t} \\
\end{align*}
✓ Solution by Mathematica
Time used: 36.077 (sec). Leaf size: 369
DSolve[(2*t+y[t]^3)+(3*t*y[t]^2+4)*D[y[t],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*}