76.16.2 problem 16
Internal
problem
ID
[17600]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
4.
Second
order
linear
equations.
Section
4.6
(Forced
vibrations,
Frequency
response,
and
Resonance).
Problems
at
page
272
Problem
number
:
16
Date
solved
:
Thursday, March 13, 2025 at 10:38:09 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{4}+2 y&=2 \cos \left (w t \right ) \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=2 \end{align*}
✓ Maple. Time used: 0.150 (sec). Leaf size: 87
ode:=diff(diff(y(t),t),t)+1/4*diff(y(t),t)+2*y(t) = 2*cos(w*t);
ic:=y(0) = 0, D(y)(0) = 2;
dsolve([ode,ic],y(t), singsol=all);
\[
y = \frac {\frac {256 \sqrt {127}\, \left (w^{4}-\frac {65}{16} w^{2}+\frac {15}{4}\right ) {\mathrm e}^{-\frac {t}{8}} \sin \left (\frac {\sqrt {127}\, t}{8}\right )}{127}+32 \,{\mathrm e}^{-\frac {t}{8}} \cos \left (\frac {\sqrt {127}\, t}{8}\right ) \left (w^{2}-2\right )-32 \cos \left (w t \right ) w^{2}+8 w \sin \left (w t \right )+64 \cos \left (w t \right )}{16 w^{4}-63 w^{2}+64}
\]
✓ Mathematica. Time used: 0.052 (sec). Leaf size: 141
ode=D[y[t],{t,2}]+1/4*D[y[t],t]+2*y[t]==2*Cos[w*t];
ic={y[0]==0,Derivative[1][y][0] == 2};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \frac {8 e^{-t/8} \left (32 \sqrt {127} w^4 \sin \left (\frac {\sqrt {127} t}{8}\right )-130 \sqrt {127} w^2 \sin \left (\frac {\sqrt {127} t}{8}\right )+508 \left (w^2-2\right ) \cos \left (\frac {\sqrt {127} t}{8}\right )-508 e^{t/8} \left (w^2-2\right ) \cos (t w)+127 e^{t/8} w \sin (t w)+120 \sqrt {127} \sin \left (\frac {\sqrt {127} t}{8}\right )\right )}{127 \left (16 w^4-63 w^2+64\right )}
\]
✓ Sympy. Time used: 0.410 (sec). Leaf size: 182
from sympy import *
t = symbols("t")
w = symbols("w")
y = Function("y")
ode = Eq(2*y(t) - 2*cos(t*w) + Derivative(y(t), t)/4 + Derivative(y(t), (t, 2)),0)
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = - \frac {32 w^{2} \cos {\left (t w \right )}}{16 w^{4} - 63 w^{2} + 64} + \frac {8 w \sin {\left (t w \right )}}{16 w^{4} - 63 w^{2} + 64} + \left (\left (\frac {32 w^{2}}{16 w^{4} - 63 w^{2} + 64} - \frac {64}{16 w^{4} - 63 w^{2} + 64}\right ) \cos {\left (\frac {\sqrt {127} t}{8} \right )} + \left (\frac {256 \sqrt {127} w^{4}}{2032 w^{4} - 8001 w^{2} + 8128} - \frac {1040 \sqrt {127} w^{2}}{2032 w^{4} - 8001 w^{2} + 8128} + \frac {960 \sqrt {127}}{2032 w^{4} - 8001 w^{2} + 8128}\right ) \sin {\left (\frac {\sqrt {127} t}{8} \right )}\right ) e^{- \frac {t}{8}} + \frac {64 \cos {\left (t w \right )}}{16 w^{4} - 63 w^{2} + 64}
\]