(a)
Let’s use linearity and \(\mathcal {L}[y\,^{\prime \prime }]=s^2\mathcal {L}[y]-sy(0)-y\,^{\prime }(0)\) to derive a formula for \(\mathcal {L}[\sin {(\omega t)}]\).
(b)
Now let’s use above and \(\mathcal {L}[y\,^{\prime }]=s\mathcal {L}[y]-y(0)\) to derive a formula for \(\mathcal {L}[\cos {(\omega t)}]\).
(c)
Let’s use \(\displaystyle \mathcal {L}(tf(t))=-\frac {dF}{ds}\) to derive formulas for \(\mathcal {L}[t\cos {(\omega t)}]\) and \(\mathcal {L}[t\sin {(\omega t)}]\).
(d)
Let’s solve the IVP \(y\,^{\prime \prime }+y=4\cos {(t)}\), \(y(0)=1\), \(y\,^{\prime }(0)=0\) using Laplace transforms.
(i)
First let’s find the Laplace transform \(Y(s)\) of the solution \(y(t)\).
(ii)
Now let’s find the inverse Laplace transform of \(Y(s)\), which is the solution \(y(t)\).