Laplace transform of a exponential. we know that Laplace

Question

Laplace transform of a exponential. we know that Laplace

Erika Bernard 2022-04-15 Answered

Laplace transform of a exponential.
we know that Laplace transform of

x^{n}

is

L [x^{n}]

provided n is a positive integer
but what is Laplace transform of

a^{x}

where a is some constant number
please help i searched on google so many times and try to solve this by its original defination that is

L [x^{n}] = \int_{0}^{\infty} a^{x} e^{- s x} d x

but how to solve further ( i am using x insted of t here)

Ask Expert 2 See Answers

You can still ask an expert for help

Want to know more about Laplace transform?

Answers (2)

Latkaxu8j

Answered 2022-04-16 Author has 11 answers

\int_{0}^{\infty} a^{x} e^{- s x} d x = \int_{0}^{\infty} e^{x \ln a} e^{- s x} d x = \int_{0}^{\infty} e^{- (s - \ln a) x} d x = \frac{1}{s - \ln a}

This is helpful 0

Drantumcem0

Answered 2022-04-17 Author has 10 answers

a^{x} e^{- s x} = {(a e^{- s})}^{x}

L {(a e^{- s})}^{x} = \int_{0}^{\infty} {(a e^{- s})}^{x} = \frac{1}{\ln (a e^{- s})} {(a e^{- s})}^{x} ∣_{0}^{\infty} = \frac{1}{\ln a - s} (0 - 1) = \frac{1}{s - \ln a}

This is helpful 0

Relevant Questions

asked 2020-12-17

Find the Laplace transform Y(s), of the solution of the IVP

y " + 3 y^{'} + 2 y = \cos (2 t)

y (0) = 0

y^{'} (0) = 1

Do not solve the IVP

See answers (1)

asked 2022-01-20

Derive the Laplace transform of the following unction: cos bt

See answers (1)

asked 2021-09-11

Consider the IVP $y y^{'} = 0$ where $y (0) = 2 and y^{'} (0) = 3$ .Solve it using Laplace transforms

See answers (1)

asked 2021-09-05

$y^{'} - \int_{0}^{t} y (τ) d τ = 4 t e^{- t}, y (0) = 3$

See answers (1)

asked 2021-10-17

A part icle moves along the curve

y = 2 \sin (π \frac{x}{2})

. As the particle passes through the point

(\frac{1}{3}, 1)

its x-coordinate increases at a rate of

\sqrt{10}

cm/s. How fast is the distance from the particle to the origin changing at this instant?

See answers (1)

asked 2022-01-21

Solve the differential equation

y^{'} = | x |, y (- 1) = 2

See answers (2)

asked 2021-02-03

The Laplace transform of

t u (t - 2) = e^{- 2 s} [\frac{s + 2 s^{2}}{s^{3}}]

True or False?

See answers (1)

Analysis

Laplace transform of a exponential. we know that Laplace

Want to know more about Laplace transform?

Answers (2)

Not exactly what you’re looking for?

Not exactly what you’re looking for?

Relevant Questions