Chapter 7: The Laplace Transform

(a) Suppose two identical pendulums are coupled by means of a spring with constant See Figure 7.R.12. Under the same assumptions made in the discussion preceding Example 3 in Section 7.6, it can be shown that when the displacement angles ${\mathbf{\theta }}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ and${\mathbf{\theta }}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ are small, the system of linear differential equations describing the motion is

$$\begin{gathered} {\mathit{\theta }}_{\mathbf{1}}^{\mathbf{*}}\mathbf{+}\frac{\mathbf{g}}{\mathbf{l}}{\mathit{\theta }}_{\mathbf{1}}\mathbf{=}\mathbf{-}\frac{\mathbf{k}}{\mathbf{m}}\left({\theta }_1-{\theta }_2\right) \\ {\mathit{\theta }}_{\mathbf{2}}^{\mathbf{*}}\mathbf{+}\frac{\mathbf{g}}{\mathbf{l}}{\mathit{\theta }}_{\mathbf{2}}\mathbf{=}\frac{\mathbf{k}}{\mathbf{m}}\left({\theta }_1-{\theta }_2\right) \end{gathered}$$

Use the Laplace transform to solve the system when ${\mathbf{\theta }}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{\theta }}_{\mathbf{0}}\mathbf{,}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{\psi }}_{\mathbf{0}}\mathbf{,}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}$ where${\mathit{\theta }}_{\mathbf{0}}$ and ${\mathit{\psi }}_{\mathbf{0}}$ are constants. For convenience let ${\mathit{\omega }}^{\mathbf{2}}\mathbf{=}\mathit{g}\mathbf{/}\mathit{l}\mathbf{,}\mathit{K}\mathbf{=}\mathit{k}\mathbf{/}\mathit{m}$.

(b) Use the solution in part (a) to discuss the motion of the coupled pendulums in the special case when the initial conditions are ${\mathbf{\theta }}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{\theta }}_{\mathbf{0}}\mathbf{,}{\mathbf{\theta }}_{\mathbf{1}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{\theta }}_{\mathbf{0}}\mathbf{,}{\mathbf{\theta }}_{\mathbf{2}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}$When the initial conditions are ${\mathbf{\theta }}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{\theta }}_{\mathbf{0}}\mathbf{,}{\mathbf{\theta }}_{\mathbf{1}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}{\mathbf{\theta }}_{\mathbf{0}}\mathbf{,}{\mathbf{\theta }}_{\mathbf{2}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$.

Use the Laplace transform to solve the given initial- value problem

$\mathbf{4}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{16}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathit{\delta }\mathbf{}\mathbf{(}\mathit{t}\mathbf{}\mathbf{-}\mathbf{}\mathbf{2}\mathbf{\pi }\mathbf{}\mathbf{)}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

In Problems 1–18 use Definition 7.1.1 to find$L\left\{f\left(t\right)\right\}.$

4.$f\left(t\right)={\{}_{0,t⩾1}^{2t+1,0⩽t<1}$

Question: In problems1-20 find either f(S) or f(t),as indicated$\mathit{L}\left\{t^{10}{e}^{-7t}\right\}$.

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

$\mathbf{}\mathbf{50}\mathbf{.}\mathbf{}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{+}\mathbf{6}\mathit{y}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{+}\mathbf{9}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{t}}\mathit{y}\mathbf{\left(}\mathit{\tau }\mathbf{\right)}\mathit{d}\mathit{\tau }\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

make up two functions f₁ and f ₂ that have the same Laplace transform. do not think profound thoughts.

In Problems 40 – 54 match the given graph with one of the functions in (a)-(f). The graph $f\left(t\right)$ 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 50

Solveequation(10)subjectto $\mathit{i}\left(0\right)\mathbf{=}\mathbf{0}$WithR, L, CandE (t)asgiven. Useagraphingutilitytographthesolutionfor$\mathbf{0}\mathbf{⩽}\mathit{t}\mathbf{⩽}\mathbf{3}$.

$$\begin{gathered} \mathbf{L}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{h}\mathbf{,}\mathbf{R}\mathbf{=}\mathbf{3}\Omega ,\mathbf{C}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{f}\mathbf{,} \\ \mathbf{E}\left(\mathbf{t}\right)\mathbf{=}\mathbf{100}\left[\mathbf{u}\left(\mathbf{t}\mathbf{-}\mathbf{1}\right)\mathbf{-}\mathbf{u}\left(\mathbf{t}\mathbf{-}\mathbf{2}\right)\right] \end{gathered}$$

Reread(iii) in the remarks on page 293find the zero-input and the zero-slate response for the IVP in problem 40.

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

1. $f\left(t\right)-f\left(t\right)u(t-a)$
   
2. $f(t-b)u(t-b)$
   
3. $f\left(t\right)u(t-a)$
   
4. $f\left(t\right)-f\left(t\right)u(t-b)$
   
5. $f\left(t\right)u(t-a)-f\left(t\right)u(t-b)$
   
6. $f(t-a)u(t-a)-f(t-a)u(t-b)$
   

Figure graph for problem 51