72.17.16 problem 18
Internal
problem
ID
[14873]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
4.
Forcing
and
Resonance.
Section
4.2
page
412
Problem
number
:
18
Date
solved
:
Thursday, March 13, 2025 at 05:16:38 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+4 y^{\prime }+20 y&=3+2 \cos \left (2 t \right ) \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 38
ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+20*y(t) = 3+2*cos(2*t);
dsolve(ode,y(t), singsol=all);
\[
y = \sin \left (4 t \right ) {\mathrm e}^{-2 t} c_{2} +\cos \left (4 t \right ) {\mathrm e}^{-2 t} c_{1} +\frac {3}{20}+\frac {\cos \left (2 t \right )}{10}+\frac {\sin \left (2 t \right )}{20}
\]
✓ Mathematica. Time used: 0.636 (sec). Leaf size: 98
ode=D[y[t],{t,2}]+4*D[y[t],t]+20*y[t]==3+2*Cos[2*t];
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to e^{-2 t} \left (\cos (4 t) \int _1^t-\frac {1}{4} e^{2 K[2]} (2 \cos (2 K[2])+3) \sin (4 K[2])dK[2]+\sin (4 t) \int _1^t\frac {1}{4} e^{2 K[1]} (2 \cos (2 K[1])+3) \cos (4 K[1])dK[1]+c_2 \cos (4 t)+c_1 \sin (4 t)\right )
\]
✓ Sympy. Time used: 3.362 (sec). Leaf size: 221
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(20*y(t) - 2*cos(2*t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 3,0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = - \frac {\left (1 - \cos {\left (2 t \right )}\right )^{3} \cos {\left (4 t \right )}}{5} - \frac {\left (1 - \cos {\left (2 t \right )}\right )^{2} \sin {\left (2 t \right )} \cos {\left (4 t \right )}}{20} + \frac {9 \left (1 - \cos {\left (2 t \right )}\right )^{2} \cos {\left (4 t \right )}}{10} + \left (C_{1} \sin {\left (4 t \right )} + C_{2} \cos {\left (4 t \right )}\right ) e^{- 2 t} + \frac {6 \sqrt {2} \sin ^{5}{\left (t \right )} \sin {\left (4 t \right )} \cos {\left (t + \frac {\pi }{4} \right )}}{5} - \frac {12 \sqrt {2} \sin ^{3}{\left (t \right )} \sin {\left (4 t \right )} \cos {\left (t + \frac {\pi }{4} \right )}}{5} - \frac {7 \sin ^{2}{\left (t \right )} \sin {\left (4 t \right )}}{5} + \frac {19 \sin {\left (t \right )} \sin {\left (4 t \right )} \cos {\left (t \right )}}{20} + \frac {\sin ^{2}{\left (3 t \right )} \sin {\left (4 t \right )}}{20} - \frac {\sin {\left (4 t \right )} \cos {\left (4 t \right )}}{8} + \frac {3 \sin {\left (4 t \right )}}{20} + \frac {89 \cos {\left (2 t \right )} \cos {\left (4 t \right )}}{80} - \frac {\cos {\left (4 t \right )} \cos {\left (6 t \right )}}{80} - \frac {17 \cos {\left (4 t \right )}}{20}
\]