Let \(k\neq 0\) be constant. Let’s solve the IVP \(y^{\,\prime \prime }+25y=50+k\delta \left (t-\frac {\pi }{5}\right )\), \(y(0)=0\), \(y^{\,\prime }(0)=1\) in terms of \(k\), and express the solution as a piecewise-defined function.

(a)
First let’s take the Laplace transform of both sides, using \(Y(s)\) to denote \(\mathcal {L}{[y(t)]}\).
(b)
Let’s solve for \(Y(s)\).
(c)
Let’s re-express \(Y(s)\) in a form conducive for taking the inverse Laplace transform.
(d)
Let’s find \(y(t)\) by taking the inverse Laplace transform of \(Y(s)\), and express the solution as a piecewise-defined function.
(e)
Let’s determine \(k\) so that the solution has the fewest terms possible when \(t\geq \pi /5\).