63.10.2 problem 7(a)

Internal problem ID [13152]
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 : Tuesday, January 28, 2025 at 05:10:44 AM
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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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