7.12.13 problem 14

Internal problem ID [395]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.6 (Forced oscillations and resonance). Problems at page 171
Problem number : 14
Date solved : Saturday, March 29, 2025 at 04:52:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+8 x^{\prime }+25 x&=200 \cos \left (t \right )+520 \sin \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=-30\\ x^{\prime }\left (0\right )&=-10 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 29

ode:=diff(diff(x(t),t),t)+8*diff(x(t),t)+25*x(t) = 200*cos(t)+520*sin(t); 
ic:=x(0) = -30, D(x)(0) = -10; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \left (-31 \cos \left (3 t \right )-52 \sin \left (3 t \right )\right ) {\mathrm e}^{-4 t}+\cos \left (t \right )+22 \sin \left (t \right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 34

ode=D[x[t],{t,2}]+8*D[x[t],t]+25*x[t]==200*Cos[t]+520*Sin[t]; 
ic={x[0]==-30,Derivative[1][x][0] ==-10}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to 22 \sin (t)-52 e^{-4 t} \sin (3 t)+\cos (t)-31 e^{-4 t} \cos (3 t) \]

✓ Sympy. Time used: 0.257 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(25*x(t) - 520*sin(t) - 200*cos(t) + 8*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): -30, Subs(Derivative(x(t), t), t, 0): -10} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (- 52 \sin {\left (3 t \right )} - 31 \cos {\left (3 t \right )}\right ) e^{- 4 t} + 22 \sin {\left (t \right )} + \cos {\left (t \right )} \]