Problem 1
Use the inverse Laplace transform formula to calculate:
(a) \(\mathcal{L}^{-1}\left[\frac{\cosh a \sqrt{s}}{s \cosh \sqrt{s}}\right],
0
Problem 1
Calculate the Laplace transform of each of the following functions: (a) \(e^{-t} \cos 2 t\) (b) \(e^{-4 t} \cosh 2 t\) (c) \(\left(t^{2}+1\right)^{2}\) (d) \(3 \cosh t-4 \sinh 5 t\) (e) \(t^{n} \sin t\)
Problem 1
Calculate the Laplace transform of $$ f(t)=\int_{0}^{t}\left(u^{2}-u+e^{-u}\right) d u . $$
Problem 2
Calculate the Laplace transform of \(f(t)=\frac{\sin t}{t}\). (Hint: First compute \(\left.\frac{d}{d s} \mathcal{L}[f](s)_{+}\right)\)
Problem 2
Solve the following integral equations: (a) \(f(t)+2 \int_{0}^{t} f(u) \cos (t-u) d u=9 e^{2 t}\) (b) \(\int_{0}^{t} f(u) d u-f^{\prime}(t)=\left\\{\begin{array}{ll}0, & 0 \leq t \leq a, \\ 1, & a \leq t,\end{array}\right\\} a>0\) (c) \(f(t)+\int_{0}^{t}(t-u) f(u) d u=\sin 2 t\) (d) \(f^{\prime \prime}(t)=\int_{0}^{t} u f(t-u) d u, \quad f(0)=-1, f^{\prime}(0)=1\) (e) \(f(t)+\int_{0}^{t} f(u) e^{-(t-u)} d u=1\) (f) \(\int_{0}^{t} f^{\prime}(u) f(t-u) d u=3 t e^{3 t}-e^{3 t}+1, \quad f(0)=0, f^{\prime}(0)>0\) (g) \(3 f^{\prime}(t)-10 f(t)+3 \int_{0}^{t} f(u) d u=10 \sin t-5, \quad f(0)=2\) (h) \(\int_{0}^{t} f(u) f(t-u) d u=2 f(t)+t-2\). (Is the solution unique?) (i) \(f^{\prime}(t)+\int_{0}^{t} f(u) d u=\sin t, \quad f(0)=1\) (j) \(\int_{0}^{t} f^{\prime \prime}(t) f(t-u) d u=t e^{a t}, \quad f(0)=\frac{1}{a}, f^{\prime}(0)=1\) (k) \(f(t)=a t+\int_{0}^{t} f(u) \sin (t-u) d u\) (l) \(f^{\prime}(t)+5 \int_{0}^{t} f(u) \cos 2(t-u) d u=10, \quad f(0)=2\) (\mathbf) \(\int_{0}^{t} f^{\prime}(u) f(t-u) d u=24 t^{3}, \quad f(0)=0\)
Problem 2
Find the Laplace transform of
$$
f(t)= \begin{cases}\sin t, & 0 \leq t \leq 2 \pi, \\ 0, & 2 \pi
Problem 2
For each \(c>0\) let \(f_{c}\) be the \(2 c\)-periodic function on the interval \([0, \infty)\) for which $$ f_{c}(t)= \begin{cases}t, & 0 \leq t \leq c, \\ 2 c-t, & c \leq t \leq 2 c .\end{cases} $$ Calculate the Laplace transform of \(f_{c}\).
Problem 2
Calculate the inverse Laplace transform of each of the following functions: (a) \(\frac{e^{-s}\left(1-e^{-s}\right)}{s\left(s^{2}+1\right)}\) (b) \(\frac{e^{-16 s}}{s\left(s^{2}+2 s+4\right)}\)
Problem 3
Evaluate the integrals (a) \(\int_{0}^{\infty} x^{2} e^{-2 x^{2}} d x\), (b) \(\int_{0}^{\infty} \sqrt[4]{x} e^{-\sqrt{x}} d x\)
Problem 3
Prove that if \(f:[0, \infty) \rightarrow \mathbb{C}\) is piecewise continuous and \(p\)-periodic \((p>0)\), then $$ \mathcal{L}[f](s)=\frac{1}{1-e^{-p s}} \int_{0}^{p} e^{-s t} f(t) d t, \quad s>0 . $$
