Chapter 8: Q12P (page 439)

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

Short Answer

Expert verified

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

Step by step solution

Given information

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

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:

$L^{- 1} (\frac{1}{(p^{2} - a^{2})}) = \cosh a L^{- 1} (\frac{a}{(p^{2} - a^{2})}) = \sinh a$

Calculate the Inverse Transformation of given function

Consider the given function.

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

Now, evaluate the inverse transformation as shown.

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

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

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

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Properties used to find the Laplace Transformation

Calculate the Inverse Transformation of given function

One App. One Place for Learning.

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Find the inverse transforms of the functionsF(p).3p+10p2-25

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Properties used to find the Laplace Transformation

Calculate the Inverse Transformation of given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Astrophysics

Particle Model of Matter

Kinematics Physics

Medical Physics

Oscillations

Study anywhere. Anytime. Across all devices.

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