14.8.7 problem 8
Internal
problem
ID
[2562]
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
:
8
Date
solved
:
Tuesday, March 04, 2025 at 02:27:50 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} 2 y^{\prime \prime }-y^{\prime }+3 y&=0 \end{align*}
With initial conditions
\begin{align*} y \left (1\right )&=1\\ y^{\prime }\left (1\right )&=1 \end{align*}
✓ Maple. Time used: 0.102 (sec). Leaf size: 70
ode:=2*diff(diff(y(t),t),t)-diff(y(t),t)+3*y(t) = 0;
ic:=y(1) = 1, D(y)(1) = 1;
dsolve([ode,ic],y(t), singsol=all);
\[
y = \frac {{\mathrm e}^{-\frac {1}{4}+\frac {t}{4}} \left (\left (\sin \left (\frac {\sqrt {23}}{4}\right ) \sqrt {23}+3 \cos \left (\frac {\sqrt {23}}{4}\right )\right ) \sin \left (\frac {\sqrt {23}\, t}{4}\right )+\left (\cos \left (\frac {\sqrt {23}}{4}\right ) \sqrt {23}-3 \sin \left (\frac {\sqrt {23}}{4}\right )\right ) \cos \left (\frac {\sqrt {23}\, t}{4}\right )\right ) \sqrt {23}}{23}
\]
✓ Mathematica. Time used: 0.032 (sec). Leaf size: 54
ode=2*D[y[t],{t,2}]-D[y[t],t]+3*y[t]==0;
ic={y[1]==1,Derivative[1][y][1] ==1};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \frac {1}{23} e^{\frac {t-1}{4}} \left (3 \sqrt {23} \sin \left (\frac {1}{4} \sqrt {23} (t-1)\right )+23 \cos \left (\frac {1}{4} \sqrt {23} (t-1)\right )\right )
\]
✓ Sympy. Time used: 0.284 (sec). Leaf size: 214
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(3*y(t) - Derivative(y(t), t) + 2*Derivative(y(t), (t, 2)),0)
ics = {y(1): 1, Subs(Derivative(y(t), t), t, 1): 1}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\left (\frac {3 \sqrt {23} \cos {\left (\frac {\sqrt {23}}{4} \right )}}{23 e^{\frac {1}{4}} \cos ^{2}{\left (\frac {\sqrt {23}}{4} \right )} + 23 e^{\frac {1}{4}} \sin ^{2}{\left (\frac {\sqrt {23}}{4} \right )}} + \frac {23 \sin {\left (\frac {\sqrt {23}}{4} \right )}}{23 e^{\frac {1}{4}} \cos ^{2}{\left (\frac {\sqrt {23}}{4} \right )} + 23 e^{\frac {1}{4}} \sin ^{2}{\left (\frac {\sqrt {23}}{4} \right )}}\right ) \sin {\left (\frac {\sqrt {23} t}{4} \right )} + \left (- \frac {3 \sqrt {23} \sin {\left (\frac {\sqrt {23}}{4} \right )}}{23 e^{\frac {1}{4}} \cos ^{2}{\left (\frac {\sqrt {23}}{4} \right )} + 23 e^{\frac {1}{4}} \sin ^{2}{\left (\frac {\sqrt {23}}{4} \right )}} + \frac {23 \cos {\left (\frac {\sqrt {23}}{4} \right )}}{23 e^{\frac {1}{4}} \cos ^{2}{\left (\frac {\sqrt {23}}{4} \right )} + 23 e^{\frac {1}{4}} \sin ^{2}{\left (\frac {\sqrt {23}}{4} \right )}}\right ) \cos {\left (\frac {\sqrt {23} t}{4} \right )}\right ) e^{\frac {t}{4}}
\]