Section 1
Definition of the Laplace Transform
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad\left\{\begin{array}{ll}-1, & t<1 \\ 1, & t \geq 1\end{array}\right.$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad\left\{\begin{array}{lr}4, & 0 \leq t<2 \\ 0, & t \geq 2\end{array}\right.$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad\left\{\begin{array}{lr}t, & 0 \leq t<1 \\ 1, & t \geq 1\end{array}\right.$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t)$$\left\{\begin{array}{lr}2 t+1, & 0 \leq t<1 \\ 0, & t \geq 1\end{array}\right.$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad\left\{\begin{array}{lr}\sin t, & 0 \leq t<\pi \\ 0, & t \geq \pi\end{array}\right.$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad\left\{\begin{array}{lr}\sin t, & 0 \leq t<\pi / 2 \\ 0, & t \geq \pi / 2\end{array}\right.$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$.
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad e^{t+7}$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad e^{-2 t-5}$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad t e^{4 t}$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad t^{2} e^{-2 t}$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad e^{-t} \sin t$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad e^{t} \cos t$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad t \cos t$
Use Definition 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad t \sin t$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad 2 t^{4}$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad t^{5}$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad 4 t-10$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad 7 t+3$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad t^{2}+6 t-3$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t)=-4 t^{2}+16 t+9$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad(t+1)^{3}$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad(2 t-1)^{3}$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad 1+e^{4 t}$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad t^{2}-e^{-9 t}+5$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad\left(1+e^{2}\right)^{2}$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad\left(e^{t}-e^{-t}\right)^{2}$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad 4 t^{2}-5 \sin 3 t$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad \cos 5 t+\sin 2 t$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad \sinh k t$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad \cosh k t$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad e^{t} \sinh t$
Use Theorem 4.1.1 to find $\mathscr{L}\{f(t)\}$. $f(t) \quad e^{-t} \cosh t$
Find $\mathscr{L}\{f(t)\}$ by first using an appropriate trigonometric identity. $f(t) \quad \sin 2 t \cos 2 t$
Find $\mathscr{L}\{f(t)\}$ by first using an appropriate trigonometric identity. $f(t) \quad \cos ^{2} t$
Find $\mathscr{L}\{f(t)\}$ by first using an appropriate trigonometric identity. $f(t) \quad \sin (4 t+5)$
Find $\mathscr{L}\{f(t)\}$ by first using an appropriate trigonometric identity. $f(t) \quad 10 \cos (t-\pi / 6)$
One definition of the gamma function $\Gamma(\alpha)$ is given by the improper integral$$\Gamma(\alpha) \quad \int_{0}^{\infty} t^{\alpha-1} e^{-t} d t, \alpha>0$$Use this definition to show that $\Gamma(\alpha+1) \quad \alpha \Gamma(\alpha)$.
Use Problem 41 to show that$$\mathscr{L}\left\{t^{\alpha}\right\} \quad \frac{\Gamma(\alpha+1)}{s^{\alpha+1}}, \alpha>-1$$This result is a generalization of Theorem 4.1.1(b).
Use the results in Problems 41 and 42 and the fact that $\Gamma\left(\frac{1}{2}\right) \quad \sqrt{\pi}$ to find the Laplace transform of the given function. $f(t) \quad t^{-1 / 2}$
Use the results in Problems 41 and 42 and the fact that $\Gamma\left(\frac{1}{2}\right) \quad \sqrt{\pi}$ to find the Laplace transform of the given function. $f(t) \quad t^{1 / 2}$
Use the results in Problems 41 and 42 and the fact that $\Gamma\left(\frac{1}{2}\right) \quad \sqrt{\pi}$ to find the Laplace transform of the given function. $f(t) \quad t^{3 / 2}$
Use the results in Problems 41 and 42 and the fact that $\Gamma\left(\frac{1}{2}\right) \quad \sqrt{\pi}$ to find the Laplace transform of the given function. $f(t) \quad 6 t^{1 / 2}-24 t^{5 / 2}$
Make up a function $F(t)$ that is of exponential order, but $f(t) \quad F^{\prime}(t)$ is not of exponential order. Make up a function $f(t)$ that is not of exponential order, but whose Laplace transform exists.
Suppose that $\mathscr{L}\left\{f_{1}(t)\right\} \quad F_{1}(s)$ for $s>c_{1}$ and that $\mathscr{L}\left\{f_{2}(t)\right\} \quad F_{2}(s)$ for $s>c_{2} .$ When does $\mathscr{L}\left\{f_{1}(t)+f_{2}(t)\right\}$$F_{1}(s)+F_{2}(s) ?$
Figure $4.1 .4$ suggests, but does not prove, that the function $f(t) \quad e^{t^{2}}$ is not of exponential order. How does the observation that $t^{2}>\ln M+c t$, for $M>0$ and $t$ sufficiently large, show that $e^{t^{2}}>M e^{c t}$ for any $c$ ?
Use part (c) of Theorem 4.1.1 to show that$$\mathscr{L}\left\{e^{(a+i b) t}\right\} \quad \frac{s-a+i b}{(s-a)^{2}+b^{2}}$$where $a$ and $b$ are real and $i^{2}=-1 .$ Show how Euler's formula (page 119 ) can then be used to deduce the results$$\mathscr{L}\left\{e^{a t} \cos b t\right\} \quad \frac{s-a}{(s-a)^{2}+b^{2}}$$and$$\mathscr{L}\left\{e^{a t} \sin b t\right\} \quad \frac{b}{(s-a)^{2}+b^{2}}$$
Under what conditions is a linear function $f(x) \quad m x+b$, $m \neq 0$, a linear transform?
The proof of part (b) of Theorem 4.1.1 requires the use of mathematical induction. Show that if$$\mathscr{L}\left\{t^{n-1}\right\} \quad(n-1) ! / s^{n}$$is assumed to be true, then $\mathscr{L}\left\{t^{n}\right\} \quad n ! / s^{n+1}$ follows.
The function $f(t) \quad 2 t e^{t^{2}} \cos e^{t^{2}}$ is not of exponential order. Nevertheless, show that the Laplace transform $\mathscr{L}\left\{2 t e^{t^{2}} \cos e^{t^{2}}\right\}$ exists. [Hint: Use integration by parts.]
If $\mathscr{L}\{f(t)\} \quad F(s)$ and $a>0$ is a constant, show that$$\mathscr{L}\{f(a t)\}=\frac{1}{a} F\left(\frac{s}{a}\right)$$This result is known as the change of scale theorem.
Use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that $a$ and $k$ are positive constants. $\mathscr{L}\left\{e^{\eta}\right\} \quad_{s-1} ; \mathscr{L}\left\{e^{a r}\right\}$
Use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that $a$ and $k$ are positive constants. $\mathscr{L}\{\cos t\} \quad \frac{s}{s^{2}+1} ; \mathscr{L}\{\cos k t\}$
Use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that $a$ and $k$ are positive constants. $\mathscr{L}\{t-\sin t\} \quad \frac{1}{s^{2}\left(s^{2}+1\right)} ; \mathscr{L}\{k t-\sin k t\}$
Use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that $a$ and $k$ are positive constants. $\mathscr{L}\{\cos t \sinh t\} \quad \frac{s^{2}-2}{s^{4}+4} ; \mathscr{L}\{\cos k t \sinh k t\}$