(a) Assume that Theorem 7.3.1 holds when the symbol a is replaced by ki, where k is a

Chapter 7, Problem 83

(choose chapter or problem)

(a) Assume that Theorem 7.3.1 holds when the symbol a is replaced by ki, where k is a real number and \(i^{2}=-1\). Show that \(\mathscr{L}\left\{t e^{k t i}\right\}\) can be used to deduce

\(\mathscr{L}\{t \cos k t\}=\frac{s^{2}-k^{2}}{\left(s^{2}+k^{2}\right)^{2}}\)

\(\mathscr{L}\{t \sin k t\}=\frac{2 k s}{\left(s^{2}+k^{2}\right)^{2}}\)

(b) Now use the Laplace transform to solve the initial value problem \(x^{\prime \prime}+\omega^{2} x=\cos \omega t\), x(0) = 0, \(x^{\prime}(0)=0\).

Text Transcription:

i^2 = -1

L{te^kti}

x^prime.prime + omega^2x = cos omega.t

x^prime(0) = 0

L{t cos kt} = s^2 - k^2/(s^2 + k^2)^2

L{t sin kt} = 2ks/(s^2 + k^2)^2