14.9.5 problem 6
Internal
problem
ID
[2571]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
2.
Second
order
differential
equations.
Section
2.2.2.
Equal
roots,
reduction
of
order.
Excercises
page
149
Problem
number
:
6
Date
solved
:
Sunday, March 30, 2025 at 12:10:30 AM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} 6 y^{\prime \prime }+2 y^{\prime }+y&=0 \end{align*}
With initial conditions
\begin{align*} y \left (2\right )&=1\\ y^{\prime }\left (2\right )&=-1 \end{align*}
✓ Maple. Time used: 0.224 (sec). Leaf size: 68
ode:=6*diff(diff(y(t),t),t)+2*diff(y(t),t)+y(t) = 0;
ic:=y(2) = 1, D(y)(2) = -1;
dsolve([ode,ic],y(t), singsol=all);
\[
y = -{\mathrm e}^{\frac {1}{3}-\frac {t}{6}} \left (\left (-\cos \left (\frac {\sqrt {5}}{3}\right )-\sin \left (\frac {\sqrt {5}}{3}\right ) \sqrt {5}\right ) \cos \left (\frac {\sqrt {5}\, t}{6}\right )+\sin \left (\frac {\sqrt {5}\, t}{6}\right ) \left (\cos \left (\frac {\sqrt {5}}{3}\right ) \sqrt {5}-\sin \left (\frac {\sqrt {5}}{3}\right )\right )\right )
\]
✓ Mathematica. Time used: 0.032 (sec). Leaf size: 51
ode=6*D[y[t],{t,2}]+2*D[y[t],t]+y[t]==0;
ic={y[2]==1,Derivative[1][y][2] ==-1};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to e^{\frac {1}{3}-\frac {t}{6}} \left (\cos \left (\frac {1}{6} \sqrt {5} (t-2)\right )-\sqrt {5} \sin \left (\frac {1}{6} \sqrt {5} (t-2)\right )\right )
\]
✓ Sympy. Time used: 0.283 (sec). Leaf size: 173
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(y(t) + 2*Derivative(y(t), t) + 6*Derivative(y(t), (t, 2)),0)
ics = {y(2): 1, Subs(Derivative(y(t), t), t, 2): -1}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\left (- \frac {\sqrt {5} e^{\frac {1}{3}} \cos {\left (\frac {\sqrt {5}}{3} \right )}}{\sin ^{2}{\left (\frac {\sqrt {5}}{3} \right )} + \cos ^{2}{\left (\frac {\sqrt {5}}{3} \right )}} + \frac {e^{\frac {1}{3}} \sin {\left (\frac {\sqrt {5}}{3} \right )}}{\sin ^{2}{\left (\frac {\sqrt {5}}{3} \right )} + \cos ^{2}{\left (\frac {\sqrt {5}}{3} \right )}}\right ) \sin {\left (\frac {\sqrt {5} t}{6} \right )} + \left (\frac {e^{\frac {1}{3}} \cos {\left (\frac {\sqrt {5}}{3} \right )}}{\sin ^{2}{\left (\frac {\sqrt {5}}{3} \right )} + \cos ^{2}{\left (\frac {\sqrt {5}}{3} \right )}} + \frac {\sqrt {5} e^{\frac {1}{3}} \sin {\left (\frac {\sqrt {5}}{3} \right )}}{\sin ^{2}{\left (\frac {\sqrt {5}}{3} \right )} + \cos ^{2}{\left (\frac {\sqrt {5}}{3} \right )}}\right ) \cos {\left (\frac {\sqrt {5} t}{6} \right )}\right ) e^{- \frac {t}{6}}
\]