Problem 3
Prove that if \(f:[0, \infty) \rightarrow \mathbb{C}\) is piecewise continuous and \(p\)-periodic \((p>0)\), then $$ \mathcal{L}[f](s)=\frac{1}{1-e^{-p s}} \int_{0}^{p} e^{-s t} f(t) d t, \quad s>0 . $$
Problem 3
For each of the following differential equations with associated initial
conditions, find a solution on the interval \([0, \infty)\).
(a) \(y^{\prime \prime}(t)+y(t)=g(t), \quad y(0)=0, \quad y^{\prime}(0)=0\),
where
$$
g(t)= \begin{cases}t, & 0 \leq t<1 \\ 1, & 1 \leq t\end{cases}
$$
(b) \(y^{\prime \prime}(t)+2 y^{\prime}(t)-3 y(t)=f(t), \quad y(0)=1, \quad
y^{\prime}(0)=0\)
where
$$
f(t)= \begin{cases}0, & 0 \leq t \leq 2 \pi, \\ \sin t, & t>2 \pi
.\end{cases}
$$
(c) \(y^{\prime \prime}(t)+2 y^{\prime}(t)+3 y(t)=\delta_{\pi}(t)+\sin t, \quad
y(0)=0, \quad y^{\prime}(0)=1\).
(d) \(y^{\prime \prime}(t)+y(t)=\delta_{\pi}(t) \cos t, \quad y(0)=0, \quad
y^{\prime}(0)=1\).
(e) \(y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)+4 y^{\prime}(t)-4
y(t)=68 e^{t} \sin 2 t\), \(y(0)=1, \quad y^{\prime}(0)=-19, \quad y^{\prime
\prime}(0)=-37 .\)
(f) \(y^{\prime \prime}(t)+9 y(t)=f_{c}(t), \quad y(0)=a, \quad
y^{\prime}(0)=b\), where for \(c>0\),
$$
f_{c}(t)= \begin{cases}0, & 0 \leq t \leq c \\ t-c, & c
Problem 4
For each of the following differential equations with associated initial conditions, find the solution on the interval \([0, \infty)\). (a) \(y^{\prime \prime}(t)+4 y^{\prime}(t)+7 y(t)=u_{1}(t), \quad y(0)=1, \quad y^{\prime}(0)=0\) (b) \(y^{\prime \prime}(t)+2 y^{\prime}(t)+3 y(t)=\delta_{\pi}(t), \quad y(0)=1, \quad y^{\prime}(0)=0\) (c) \(y^{\prime \prime}(t)-2 y^{\prime}(t)+y(t)=(-1)^{[t]}, \quad y(0)=0, \quad y^{\prime}(0)=0\) (d) \(y^{\prime \prime}(t)+2 y^{\prime}(t)+2 y(t)=\delta_{\pi}(t), \quad y(0)=1, \quad y^{\prime}(0)=0\) (e) \(y^{\prime \prime}(t)+4 y(t)=\delta_{\pi}(t)-\delta_{2 \pi}(t), \quad y(0)=0, \quad y^{\prime}(0)=0\) (f) \(y^{\prime \prime \prime}(t)-y(t)=\left\\{\begin{array}{ll}1, & \pi \leq t \leq 2 \pi, \\ 0, & \text { otherwise, }\end{array}\right\\} \quad y(0)=y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=1\)
Problem 4
Prove, for every \(p>0, \quad \int_{0}^{1}\left(\ln \frac{1}{x}\right)^{p-1} d x=\Gamma(p)\).
Problem 4
Let \(f:[0, \infty) \rightarrow \mathbb{R}\) be a 4-periodic function such that $$ f(t)=\left\\{\begin{array}{rc} 1, & 0 \leq t<2 \\ -1, & 2 \leq t<4 \end{array}\right. $$ Prove that $$ \mathcal{L}[f](s)=\frac{\tanh s}{s}, \quad s>0 . $$
Problem 4
Let \(y_{1}(t)\) be the solution of $$ y^{\prime \prime}(t)-4 y^{\prime}(t)+4 y(t)=f_{1}(t), \quad y(0)=1, \quad y^{\prime}(0)=1, \quad t \geq 0, $$ and let \(y_{2}(t)\) be the solution of $$ y^{\prime \prime}(t)-4 y^{\prime}(t)+4 y(t)=f_{2}(t), \quad y(0)=1, \quad y^{\prime}(0)=1, \quad t \geq 0 . $$ Suppose that \(f_{1}(t) \leq f_{2}(t)\), for each \(t \geq 0\), and that \(f_{1}\) and \(f_{2}\) are continuous and bounded on the interval \([0, \infty)\). Prove that \(y_{1}(t) \leq y_{2}(t)\) for each \(t \geq 0\).
Problem 4
Prove each of the following formulae: (a) \(\mathcal{L}\left[\sin ^{2} t\right](s)=\frac{2}{s\left(s^{2}+4\right)}\) (b) \(\mathcal{L}[A \cos (\omega t+\theta)](s)=\frac{A(s \cos \theta-\omega \sin \theta)}{s^{2}+\omega^{2}}\) (c) \(\mathcal{L}[\cos a t \cosh a t](s)=\frac{s^{3}}{s^{4}+4 a^{4}}\) (d) \(\mathcal{L}\left[\left(t^{2}-5 t+6\right) e^{2 t}\right]=\frac{6 s^{2}-29 s+36}{(s-2)^{3}}\)
Problem 5
Prove, for every \(p, q>-1, \quad \int_{0}^{1} x^{p}\left(\ln \frac{1}{x}\right)^{q} d x=\frac{\Gamma(q+1)}{(p+1)^{q+1}}\)
Problem 5
By using the definition and properties of the Laplace transform, compute each of the following integrals: (a) \(\int_{0}^{\infty} t e^{-2 t} \cos t d t\) (b) \(\int_{0}^{\infty} t^{3} e^{-t} \sin t d t\) (c) \(\int_{0}^{\infty} x^{4} e^{-x} d x\) (d) \(\int_{0}^{\infty} x^{6} e^{-3 x} d x\)
Problem 5
Solve the finite differential-difference equation \(y^{\prime}(t)+y(t-1)=t^{2}, t>0\), where \(y(t)=0\) for every \(t \leq 0\). (Hint: Use power series.)
