1.
Calculate the cosine transform of \(e^{-x^{2}}\text{.}\)
Answer.
\(\frac{\sqrt{\pi}}{2}e^{-s^{2}/4}\text{.}\)
Exercises 6.7 Exercise
2.
Find cosine transform of function \(f(x)\) such that
\begin{equation*}
f(x)=\begin{cases}
\cos x, & \text{when} \quad 0 \lt x \lt a\\
0, & \text{when} \quad x \gt a
\end{cases}
\end{equation*}
Answer.
\begin{equation*}
\frac{1}{2}\left[\frac{\sin(1+s)a}{1+s}+\frac{\sin(1-s)a}{1-s}\right]
\end{equation*}
3.
Calculate the cosine and sine transforms of \(e^{-bx}\text{,}\) where b is a positive integer.
Answer.
\(\frac{b}{b^{2}+s^{2}} \) and \(\frac{s}{s^{2}+b^{2}}\)
4.
Find the Fourier complex transform of \(f(x)\text{,}\) if
\begin{equation*}
f(x)=\begin{cases}
e^{-i\omega x}, & \text{when} \quad a \lt x \lt b\\
0, & \text{when} \quad x \lt a, x \gt b
\end{cases}
\end{equation*}
Answer.
\begin{equation*}
i\left[\frac{e^{-i(s_\omega)b}-e^{-i(s_\omega)a}}{s+\omega}\right]
\end{equation*}
5.
Find \(\mathscr{L}\{t\cos at\}\)
Answer.
\begin{equation*}
\frac{s^{2}-a^{2}}{(s^{2}-a^{2})^{2}}.
\end{equation*}
6.
Evaluate
\begin{equation*}
\mathscr{L}\{\frac{\sin t}{t}\}
\end{equation*}
Answer.
\begin{equation*}
\cot^{-1}s.
\end{equation*}
7.
Find the Laplace Transform of \(f(t)\) defined as
\begin{equation*}
f(t) =\begin{cases}
t+1, \quad 0 \leq t \leq 2\\
3, \quad t \gt 2
\end{cases}
\end{equation*}
Answer.
\begin{equation*}
\frac{1}{s^{2}}[s+1-e^{-2s}].
\end{equation*}
8.
Find
\begin{equation*}
\mathscr{L}^{-1}\{\frac{1}{s^{4}}\}
\end{equation*}
Answer.
\begin{equation*}
\frac{t^{3}}{6}.
\end{equation*}
9.
Find inverse Laplace transform of \(\frac{3s+7}{s^{2}-2s-3}\text{.}\)
Answer.
\(4e^{3t}-e^{-t}.\)
10.
Evaluate
\begin{equation*}
\mathscr{L}^{-1}\{\frac{2s-1}{s^{2}-s}\}
\end{equation*}
Answer.
\begin{equation*}
1-\frac{3}{2}e^{-t}+\frac{1}{2}e^{t}.
\end{equation*}
11.
Use convolution theorem to find
\begin{equation*}
\mathscr{L}^{-1}\{\frac{s^{2}}{(s^{2}+4)^{2}}\}
\end{equation*}
Answer.
\begin{equation*}
\frac{1}{2}\left[t\cos 2t+\frac{1}{2}\sin 2t\right].
\end{equation*}
12.
Using Laplace transform, solve the following differential equation
\begin{equation*}
y''+2y'+2y = 5\sin t
\end{equation*}
where \(y(0)=y'(0)=0\) at \(t=0\text{.}\)
Answer.
\begin{equation*}
e^{-t}[\cos t+\sin t]-2\cos t+\sin t.
\end{equation*}
13.
Prove that
\begin{equation*}
\int\limits_{0}^{\infty}e^{-x^{2}}\,dx = \frac{\pi}{2}.
\end{equation*}
