63.10.2 problem 7(a)

Internal problem ID [13073]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.3.2 Resonance Exercises page 114
Problem number : 7(a)
Date solved : Wednesday, March 05, 2025 at 09:16:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+w^{2} x&=\cos \left (\beta t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 27

ode:=diff(diff(x(t),t),t)+w^2*x(t) = cos(beta*t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {\cos \left (t w \right )-\cos \left (\beta t \right )}{\beta ^{2}-w^{2}} \]

✓ Mathematica. Time used: 0.105 (sec). Leaf size: 111

ode=D[x[t],{t,2}]+w^2*x[t]==Cos[\[Beta]*t]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to -\sin (t w) \int _1^0\frac {\cos (w K[2]) \cos (\beta K[2])}{w}dK[2]+\sin (t w) \int _1^t\frac {\cos (w K[2]) \cos (\beta K[2])}{w}dK[2]+\cos (t w) \left (\int _1^t-\frac {\cos (\beta K[1]) \sin (w K[1])}{w}dK[1]-\int _1^0-\frac {\cos (\beta K[1]) \sin (w K[1])}{w}dK[1]\right ) \]

✓ Sympy. Time used: 0.140 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
BETA = symbols("BETA") 
w = symbols("w") 
x = Function("x") 
ode = Eq(w**2*x(t) - cos(BETA*t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {\cos {\left (\beta t \right )}}{- \beta ^{2} + w^{2}} - \frac {e^{i t w}}{- 2 \beta ^{2} + 2 w^{2}} - \frac {e^{- i t w}}{- 2 \beta ^{2} + 2 w^{2}} \]