14.8.8 problem 9
Internal
problem
ID
[2563]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
2.
Second
order
differential
equations.
Section
2.2.1
Linear
equations
with
constant
coefficients
(complex
roots).
Excercises
page
144
Problem
number
:
9
Date
solved
:
Tuesday, March 04, 2025 at 02:27:55 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} 3 y^{\prime \prime }-2 y^{\prime }+4 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: 0.113 (sec). Leaf size: 70
ode:=3*diff(diff(y(t),t),t)-2*diff(y(t),t)+4*y(t) = 0;
ic:=y(2) = 1, D(y)(2) = -1;
dsolve([ode,ic],y(t), singsol=all);
\[
y = \frac {{\mathrm e}^{-\frac {2}{3}+\frac {t}{3}} \left (\left (\sin \left (\frac {2 \sqrt {11}}{3}\right ) \sqrt {11}-4 \cos \left (\frac {2 \sqrt {11}}{3}\right )\right ) \sin \left (\frac {\sqrt {11}\, t}{3}\right )+\left (\cos \left (\frac {2 \sqrt {11}}{3}\right ) \sqrt {11}+4 \sin \left (\frac {2 \sqrt {11}}{3}\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{3}\right )\right ) \sqrt {11}}{11}
\]
✓ Mathematica. Time used: 0.032 (sec). Leaf size: 54
ode=3*D[y[t],{t,2}]-2*D[y[t],t]+4*y[t]==0;
ic={y[2]==1,Derivative[1][y][2] ==-1};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \frac {1}{11} e^{\frac {t-2}{3}} \left (11 \cos \left (\frac {1}{3} \sqrt {11} (t-2)\right )-4 \sqrt {11} \sin \left (\frac {1}{3} \sqrt {11} (t-2)\right )\right )
\]
✓ Sympy. Time used: 0.280 (sec). Leaf size: 235
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(4*y(t) - 2*Derivative(y(t), t) + 3*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)
\[
y{\left (t \right )} = \left (\left (- \frac {4 \sqrt {11} \cos {\left (\frac {2 \sqrt {11}}{3} \right )}}{11 e^{\frac {2}{3}} \cos ^{2}{\left (\frac {2 \sqrt {11}}{3} \right )} + 11 e^{\frac {2}{3}} \sin ^{2}{\left (\frac {2 \sqrt {11}}{3} \right )}} + \frac {11 \sin {\left (\frac {2 \sqrt {11}}{3} \right )}}{11 e^{\frac {2}{3}} \cos ^{2}{\left (\frac {2 \sqrt {11}}{3} \right )} + 11 e^{\frac {2}{3}} \sin ^{2}{\left (\frac {2 \sqrt {11}}{3} \right )}}\right ) \sin {\left (\frac {\sqrt {11} t}{3} \right )} + \left (\frac {11 \cos {\left (\frac {2 \sqrt {11}}{3} \right )}}{11 e^{\frac {2}{3}} \cos ^{2}{\left (\frac {2 \sqrt {11}}{3} \right )} + 11 e^{\frac {2}{3}} \sin ^{2}{\left (\frac {2 \sqrt {11}}{3} \right )}} + \frac {4 \sqrt {11} \sin {\left (\frac {2 \sqrt {11}}{3} \right )}}{11 e^{\frac {2}{3}} \cos ^{2}{\left (\frac {2 \sqrt {11}}{3} \right )} + 11 e^{\frac {2}{3}} \sin ^{2}{\left (\frac {2 \sqrt {11}}{3} \right )}}\right ) \cos {\left (\frac {\sqrt {11} t}{3} \right )}\right ) e^{\frac {t}{3}}
\]