34E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Theorem 6 in Section 7.3 on page 364 can be expressed in terms of the inverse Laplace transform as

${\mathcal{L}}^{-1}\left\{\frac{d^{n}F}{d{s}^n}\right\}\left(t\right)={\left(-t\right)}^{n}f\left(t\right),$

Where $f={\mathcal{L}}^{-1}\left\{F\right\}$.Use this equation in Problems 33-36 to compute ${\mathcal{L}}^{-1}\left\{F\right\}$.$F\left(s\right)=ln\left(\frac{s-4}{s-3}\right)$.

Step-by-Step Solution

Answer

${\mathcal{L}}^{-1}\left(F\right)=\frac{1}{t}\left(e^{3t}-e^{4t}\right)$

Step 1: Simplify the function and find derivative

Using the property of function  $\ln t$  we get:

$$\begin{gathered} F\left(s\right)=\ln \left(\frac{s-4}{s-3}\right) \\ F\left(s\right)=\ln \left(s-4\right)-\ln \left(s-3\right) \end{gathered}$$

Find the derivative of F with respect to s:

$$\begin{gathered} \frac{dF}{ds}=\frac{d}{ds}\left(\ln \left(s-4\right)-\ln \left(s-3\right)\right) \\ \frac{dF}{ds}=\frac{d}{ds}\left(\ln \left(s-4\right)\right)-\frac{d}{ds}\left(\ln \left(s-3\right)\right) \\ \frac{dF}{ds}=\frac{1}{s-4}-\frac{1}{s-3} \end{gathered}$$

Step 2: Find the Laplace inverse

From the given condition, we have $$\begin{gathered} {\mathcal{L}}^{-1}\left(F\right)=\frac{1}{{\left(-t\right)}^1}{\mathcal{L}}^{-1} \left\{\frac{dF}{ds}\right\} \\ {\mathcal{L}}^{-1}\left(F\right)=\frac{1}{\left(-t\right)}{\mathcal{L}}^{-1} \left(\frac{1}{s-4}-\frac{1}{s-3}\right) \\ {\mathcal{L}}^{-1}\left(F\right)=\frac{1}{\left(t\right)}\left(-{\mathcal{L}}^{-1} \left(\frac{1}{s-4}\right)+ \left(\frac{1}{s-3}\right)\right) \\ {\mathcal{L}}^{-1}\left(F\right)=\frac{1}{\left(t\right)}\left(e^{3t^{}}- e^{4t}\right) \\ Therefore, \\ {\mathcal{L}}^{-1}\left(F\right)=\frac{1}{\left(t\right)}\left(e^{3t^{}}- e^{4t}\right) \end{gathered}$$