Q7.3 - 11E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]

$\mathbf{coshbt}$

Step-by-Step Solution

Answer

The Laplace transform for the given equation is $\left[\frac{s}{s^2-b^2}\right]$.

Definition of Laplace transform

• The integral transform of a given derivative function with real variable t into a complex function with variable s is known as the Laplace transform. 
• Let f(t) be supplied for t(0), and assume that the function meets certain constraints that will be presented subsequently.
• The Laplace transform formula defines the Laplace transform of f(t), which is indicated by $\mathcal{L}\left\{f\left(t\right)\right\}$ or F(s).

Determine the Laplace transform for the given equation

Given that $\cosh bt$.

Find the Laplace transform of $\cosh bt$ using $coshax=\frac{1}{2}\left[e^{ax}-e^{-ax}\right]$ , $\mathcal{L}\left\{a{f}_1+b{f}_2\right\}=a\mathcal{L}\left\{f_1\right\}+b\mathcal{L}\left\{f_2\right\}$, $\mathcal{L}\left\{e^{at}\right\}=\frac{1}{s-a}$, $\frac{c}{a}\pm \frac{d}{b}=\frac{bc\pm cd}{ab}$ and $(a+b)(a-b)=a^2-b^2$ as follows:

$\begin{array}{rcl}\mathcal{L}\left\{\cosh bt\right\}& =& \frac{1}{2}\left[e^{bt}-e^{-bt}\right]\\ & =& \frac{1}{2}\left[\mathcal{L}\left\{e^{bt}\right\}+\mathcal{L}\left\{e^{-bt}\right\}\right]\\ & =& \frac{1}{2}\left[\frac{1}{s-b}+\frac{1}{s-(-b)}\right]\\ & =& \frac{1}{2}\left[\frac{1}{s-b}+\frac{1}{s+b}\right]\end{array}$

Simplify the equation as:

$\begin{array}{rcl}\mathcal{L}\left\{\cosh bt\right\}& =& \frac{1}{2}\left[\frac{s+b+s-b}{(s-b)(s+b)}\right]\\ & =& \frac{1}{2}\left[\frac{2s}{s^2-b^2}\right]\\ & =& \left[\frac{s}{s^2-b^2}\right]\end{array}$

Therefore, the Laplace transform for the given equation is $\left[\frac{s}{s^2-b^2}\right]$.