68.6.14 problem 15
Internal
problem
ID
[17324]
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
:
Thursday, October 02, 2025 at 02:02:34 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} 2 t +y^{3}+\left (3 t y^{2}+4\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 390
ode:=2*t+y(t)^3+(3*t*y(t)^2+4)*diff(y(t),t) = 0;
dsolve(ode,y(t), singsol=all);
\begin{align*}
y &= -\frac {2 \left (12^{{1}/{3}} t -\frac {{\left (-9 \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 ) t^{2}\right )}^{{2}/{3}}}{4}\right ) 12^{{1}/{3}}}{3 {\left (-9 \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 ) t^{2}\right )}^{{1}/{3}} t} \\
y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (4 i 2^{{2}/{3}} 3^{{5}/{6}} t +i {\left (-9 \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 ) t^{2}\right )}^{{2}/{3}} \sqrt {3}-4 \,2^{{2}/{3}} 3^{{1}/{3}} t +{\left (-9 \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 ) t^{2}\right )}^{{2}/{3}}\right )}{12 {\left (-9 \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 ) t^{2}\right )}^{{1}/{3}} t} \\
y &= \frac {3^{{1}/{3}} 2^{{2}/{3}} \left (4 \,2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) t +\left (i \sqrt {3}-1\right ) {\left (-9 \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 ) t^{2}\right )}^{{2}/{3}}\right )}{12 {\left (-9 \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 ) t^{2}\right )}^{{1}/{3}} t} \\
\end{align*}
✓ Mathematica. Time used: 45.194 (sec). Leaf size: 369
ode=(2*t+y[t]^3)+(3*t*y[t]^2+4)*D[y[t],t]==0;
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(2*t + (3*t*y(t)**2 + 4)*Derivative(y(t), t) + y(t)**3,0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
Timed Out