Exercises

(1)
Suppose \(f(t)\) is a continuous function on \([0,\infty )\) and \(\mathcal {L}(f(t))=\frac {5}{s^2+1}\). Which is the Laplace transform of the solution to \(y\,^{\prime }-2y=f(t)\), \(y(0)=0\)?
(a)
\(\dfrac {5s-10}{s^2+1}\)
(b)
\(\dfrac {5}{s^3-s^2-2}\)
(c)
\(\dfrac {5}{s^3-2s^2+s-2}\)
(d)
\(\dfrac {5}{s^3-s^2+2}\)
(e)
\(\dfrac {5}{s^3-2s^2-s+2}\)
(f)
\(\dfrac {5}{s^3-s^2-2s-2}\)
(2)
Which is the Laplace transform of the solution to the IVP
\(y^{\,\prime \prime }-y^{\,\prime }-2y=27te^{-t}\), \(y(0)=0\), \(y^{\,\prime }(0)=0\)?
(a)
\(\dfrac {27}{(s+1)^2}\)
(b)
\(\dfrac {27}{(s+1)^2(s-2)}\)
(c)
\(\dfrac {27s}{(s+1)^2(s-2)}\)
(d)
\(\dfrac {27}{(s+1)^3(s-2)}\)
(e)
\(\dfrac {27s}{(s+1)^3(s-2)}\)
(f)
\(\dfrac {27s^2}{(s+1)^3(s-2)}\)
(3)
Determine \(y(1)\) given that \(y(t)\) satisfies the IVP in the previous exercise. Round to the nearest hundredth.
(4)
Suppose that \(f(t)\) is a differentiable function on \([0,\infty )\) and that \(f^{\,\prime }(t)\) is exponential order \(s_0\) for some real number \(s_0\). Determine
\[\displaystyle \lim _{s\rightarrow \infty } s\mathcal {L}(f(t))-f(0).\]

Hints: \(\displaystyle s\mathcal {L}(f(t))-f(0)=\mathcal {L}(f\,^{\prime }(t))=\int _0^{\infty } f^{\,\prime }(t)e^{-st}\,\mathrm {d}t\), and since \(f^{\,\prime }(t)\) is of exponential order \(s_0\), the limit as \(s\rightarrow \infty \) of the improper integral is equal to the improper integral of the limit of the integrand.

(5)
Let \(a,b,c\) be real numbers and assume \(a\neq 0\). Let \(y(t)=\mathcal {L}^{-1}\left (\dfrac {as+b}{as^2+bs+c}\right )\). Determine \(y(0)\).

Hint: use the result of the previous exercise.

(a)
\(0\)
(b)
\(1\)
(c)
\(2\)
(d)
\(a\)
(e)
\(1/a\)
(f)
\(b\)
(g)
\(b/a\)
(h)
None of these