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Chapter 8: Q19P (page 429)

Prove the general formula L29.

Short Answer

Expert verified

Answer

The function $L (e^{- a t} g (f)) = G (p + a)$ is proved

Step by step solution

01

Given information

The given function which has to prove is $L (f) = \int_{0} f (t) e^{- p t} d t = F (p)$ and function which has to prove is $L (e^{- a t} g (f)) = G (p + a)$

02

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)

03

Proof for the given function

It can be calculated as,

$L (f) = \int_{0}^{\infty} f (t) e^{- p t} d t L (e^{- a t} g (t)) = \int_{0}^{p} e^{- a t} e^{- p t} g (t) d t L (e^{- a t} g (t)) = \int_{0}^{p} e^{- a t} e^{- w - μ t} g (t) d t L (e^{- a t} g (t)) = \int_{0}^{\infty} e^{- (p + u) □} g (t) d t$

Again, with use of

$\int_{0}^{"} f (t) e^{- p} d t = F (p)$

$L (e^{- 2} (t))$ Can be calculated as,

$L (e^{- a t} g (t)) = \int_{0} e^{- (p + a)} g (t) d t = G (p + a)$

Hence, $L (e^{- a t} g (t)) = G (p + a)$ .

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Most popular questions from this chapter

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$y^{"} - 6 y' + 9 y = t e^{3 t}$ ,

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$y " - 4 y' + 4 y = 6 e^{2 t}, y_{0} = y^{'} = 0$

Solve Example 4 using the general solution $y = a \sinh x + b \cosh x$ .

a) Show that $\nabla \cdot (e_{r} / r^{2}) = 0$ for $r > 0$ .

(b) Show that $\nabla (1 / r) = - e_{r} / r^{2}$ .

Suppose the rate at which bacteria in a culture grow is proportional to the number present at any time. Write and solve the differential equation for the number N of bacteria as a function of time t if there are $N_{0}$ bacteria when $t = 0$ . Again note that (except for a change of sign) this is the same differential equation and solution as in the preceding problems.

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