72.1.15 problem 18
Internal
problem
ID
[14544]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
1.
First-Order
Differential
Equations.
Exercises
section
1.2.
page
33
Problem
number
:
18
Date
solved
:
Monday, March 31, 2025 at 12:30:23 PM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {4 t}{1+3 y^{2}} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 278
ode:=diff(y(t),t) = 4*t/(1+3*y(t)^2);
dsolve(ode,y(t), singsol=all);
\begin{align*}
y &= \frac {\left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{2}/{3}}-3}{3 \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{2}/{3}}+3 i \sqrt {3}-3}{6 \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{2}/{3}}+3 i \sqrt {3}-\left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{2}/{3}}+3}{6 \left (27 t^{2}+54 c_1 +3 \sqrt {81 t^{4}+324 c_1 \,t^{2}+324 c_1^{2}+3}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 2.315 (sec). Leaf size: 298
ode=D[y[t],t]==4*t/(1+3*y[t]^2);
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*}
y(t)\to \frac {\sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2}}{\sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}} \\
y(t)\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}} \\
y(t)\to \frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 t^2+\sqrt {108+729 \left (2 t^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}} \\
\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-4*t/(3*y(t)**2 + 1) + Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
Timed Out