Question

Find the inverse transforms of the functions$F\left(p\right)$.$\frac{3p+10}{p^2-25}$

Short answer

The inverse transform of function$\frac{3p+10}{p^2-25}$ is$L^{-1}\left(\frac{3p+10}{p^2-25}\right)=3\cosh \left(5t\right)+2\sinh \left(5t\right)$

Step-by-step solution

Given information

The given function is$\frac{3p+10}{p^2-25}$

Definition of Laplace Transformation

A transformation of a function f(x) into the function g(t) that is useful especially in reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation.

The inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t)

Properties used to find the Laplace Transformation

These properties are used to solve the function:

$$\begin{gathered} L^{-1}\left(\frac{1}{\left(p^2-a^2\right)}\right)=\cosh a \\ {L}^{-1}\left(\frac{a}{\left(p^2-a^2\right)}\right)=\sinh a \end{gathered}$$

Calculate the Inverse Transformation of given function

Consider the given function.

$F\left(p\right)=\frac{3p+10}{p^2-25}$

Now, evaluate the inverse transformation as shown.

$$\begin{gathered} L^{-1}\left(\frac{3p+10}{p^2-25}\right)=L^{-1}\left(\frac{3p}{p^2-25}\right)+L^{-1}\left(\frac{10}{p^2-25}\right) \\ =L^{-1}\left(\frac{3p}{p^2-5^2}\right)+L^{-1}\left(\frac{2\times 5}{p^2-5^2}\right) \\ =3L^{-1}\left(\frac{p}{p^2-5^2}\right)+2L^{-1}\left(\frac{5}{p^2-5^2}\right) \\ =3\cosh \left(5t\right)+2\sinh \left(5t\right) \end{gathered}$$

Hence$L^{-1}\left(\frac{3p+10}{p^2-25}\right)=3\cosh \left(5t\right)+2\sinh \left(5t\right)$