Chapter 7: The Laplace Transform

In problems 47and 48use one of the inverses Laplace transform found in problems 31-54to solve the given initial-value problems.

${\mathit{y}}^{\mathbf{n}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{=}\mathbf{10}\mathit{c}\mathit{o}\mathit{s}\mathbf{5}\mathit{t}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{0}$

A uniform cantilever beam of length L is embedded at its left end (x=0) and free at its right end. Find the deflection y(x) if the load per unit length is given by

$\mathbf{w}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{2}{\mathbf{w}}_{\mathbf{0}}}{\mathbf{L}}\left[\frac{L}{2}-x+\left(x-\frac{L}{2}\right)u\left(x-\frac{L}{2}\right)\right]$.

Use the Laplace transform to solve the given integral equation or integraodifferential equation.: $t-2f\left(t\right)=\underset{0}{\overset{t}{\int }}(e^{\tau }-e^{-\tau })f(t-\tau )d\tau$

Figure 7.1.4 suggests, but does not prove, that the function$f\left(t\right)=e^{t^2}$ is not of exponential order. How does the observation that$t^2>lnM+ct,$ for$M>0$ and t sufficiently large, show that $e^{t^2}>M{e}^{ct}$for any c?

In Problems$37-48$find either $F\left(s\right)$or$f\left(t\right)$, as indicated.$L^{-1}\left\{\frac{e^{-2s}}{s^2(s-1)}\right\}$

InproblemusetheLaplacetransformand these inversestosolvethegiveninitial-valueproblems.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{4}\mathit{t}\mathbf{,}\mathit{y}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{0}$

When a uniform beam is supported by an elastic foundation, the differential equation for its deflection y(x) is

$\mathbf{E}\mathbf{I}\frac{{\mathbf{d}}^{\mathbf{4}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{4}}}\mathbf{+}\mathbf{k}\mathbf{y}\mathbf{=}\mathbf{w}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{,}$

where k is the modulus of the foundation and -ky is the restoring force of the foundation that acts in the direction opposite to that of the load w(x). See Figure 7.R.11. For algebraic convenience suppose that the differential equation is written as

$\frac{{\mathbf{d}}^{\mathbf{4}}\mathbf{y}}{\mathbf{d}{\mathbf{x}}^{\mathbf{4}}}\mathbf{+}\mathbf{4}{\mathit{a}}^{\mathbf{4}}\mathit{y}\mathbf{=}\frac{\mathbf{w}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}{\mathbf{E}\mathbf{I}}\mathbf{.}$

where$\mathit{a}\mathbf{=}\mathbf{(}\mathbf{k}\mathbf{/}\mathbf{4}\mathbf{E}\mathbf{I}{\mathbf{)}}^{\mathbf{1}\mathbf{/}\mathbf{4}}\mathbf{.}$ Assume$\mathit{L}\mathbf{=}\mathit{\pi }$ and a=1 Find the deflection y(x) of a beam that is supported on an elastic foundation when

(a) the beam is simply supported at both ends and a constant load W₀is uniformly distributed along its length,

(b) the beam is embedded at both ends and w(x). is a concentrated load W₀applied at $\mathit{x}\mathbf{=}\mathit{\pi }\mathbf{/}\mathbf{2}\mathbf{.}$[Hint: In both parts of this problem use the table of Laplace transforms in Appendix C and the fact that ${\mathit{s}}^{\mathbf{4}}\mathbf{+}\mathbf{4}\mathbf{=}\left(s^2-2s+2\right)\left(s^2+2s+2\right)$.

Use the Laplace transform to solve the given integral equation or integrodifferential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{sint}\mathbf{-}\underset{\mathbf{0}}{\overset{\mathbf{t}}{\int }}\mathbf{y}\mathbf{\left(}\mathbf{\tau }\mathbf{\right)}\mathbf{d\tau }\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

In Problems 40 – 54 match the given graph with one of the functions in (a)-(f). The graph is given is Figure 7.3.11

$$\begin{gathered} a.f\left(t\right)-f\left(t\right)u(t-a) \\ b.f(t-b)u(t-b) \\ c.f\left(t\right)u(t-a) \\ d.f\left(t\right)-f\left(t\right)u(t-b) \\ e.f\left(t\right)u(t-a)-f\left(t\right)u(t-b) \\ f.f(t-a)u(t-a)-f(t-a)u(t-b) \end{gathered}$$

Figure graph for problem 49

a)With a slight changes in notation the transform in(6) is the same as $\mathit{L}\left\{f\text{'}\left(t\right)\right\}\mathbf{=}\mathit{s}\mathit{L}\left\{f\left(t\right)-f\left(0\right)\right\}$with $\mathit{f}\left(t\right)\mathbf{=}\mathit{a}\mathit{t}{\mathit{e}}^{\mathbf{a}\mathbf{t}}$,discuss how this result in conjunction with ©of Theorem can 7.11 can be used to evaluate localid="1663977247068" $\mathit{L}\mathbf{\left\{}{\mathbf{te}}^{\mathbf{at}}\mathbf{\right\}}$

b)proceed as in part (a).but this time discuss how to use (7) with $\mathit{f}\left(t\right)\mathbf{=}\mathit{t}\mathit{s}\mathit{i}\mathit{n}\mathit{k}\mathit{t}$ in conjunction with(d) and(e)of theorem to evaluate $\mathit{L}\left\{tsinkt\right\}\mathbf{.}$