Question

This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of \(F\) is the set of all real numbers s such that the improper integral converges. Find the Laplace transform \(F\) of each of the following functions and give the domain of \(F\).\(f(x)=e^{a x}\)

Short answer

The Laplace transform of \( e^{ax} \) is \( \frac{1}{s-a} \) for \( s > a \).

Step-by-step solution

Understand the Laplace Transform Definition

The Laplace transform of a function \( f(t) \) is defined as \( \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt \). We need to apply this definition to the function \( f(t) = e^{at} \).

Set Up the Integral

Substitute \( f(t) = e^{at} \) into the Laplace transform definition to get:\[ \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} e^{at} \, dt = \int_{0}^{\infty} e^{(a-s)t} \, dt \].

Evaluate the Integral

The integral becomes \( \int_{0}^{\infty} e^{(a-s)t} \, dt \). To solve this, we integrate to find:\[ \int e^{(a-s)t} \, dt = \frac{1}{a-s} e^{(a-s)t} + C \].

Apply the Limits of Integration

Evaluate the definite integral from 0 to \( \infty \):\[ \left[ \frac{1}{a-s} e^{(a-s)t} \right]_{0}^{\infty} = \left( \lim_{t \to \infty} \frac{1}{a-s} e^{(a-s)t} \right) - \left( \frac{1}{a-s} e^{0} \right) \].

Determine the Condition for Convergence

For the integral to converge, \( a-s < 0 \) or \( s > a \). Under this condition, the limit as \( t \to \infty \) will be 0, since \( e^{(a-s)t} \to 0 \).

Simplify the Expression

With \( s > a \), the evaluated integral becomes:\[ 0 - \frac{1}{a-s} = - \frac{1}{a-s} = \frac{1}{s-a} \]. Thus, the Laplace transform \( F(s) = \frac{1}{s-a} \) is valid for \( s > a \).

State the Domain of the Laplace Transform

The domain of the Laplace transform \( \mathcal{L}\{e^{at}\} \) is all \( s > a \) to ensure convergence of the improper integral.

Key concepts

Initial-Value Problems

Initial-value problems are a fundamental concept in differential equations, often used to predict the behavior of dynamic systems. These problems typically involve finding a function that satisfies a differential equation and certain specified values, known as initial conditions, at a starting point. Initial conditions are crucial as they determine the specific solution among many possibilities. For example, imagine you have a differential equation describing how temperature changes over time. The initial value might represent the starting temperature at time zero. By integrating these initial conditions into the differential equation, you can derive a unique solution that describes how the temperature evolves.

Differential Equations

Differential equations form the backbone of mathematical modelling for phenomena involving rates of change. Essentially, they are equations involving an unknown function and its derivatives. They can be classified into ordinary differential equations (ODEs) or partial differential equations (PDEs), depending on whether they involve ordinary derivatives or partial derivatives, respectively. Solving differential equations often requires finding a function that satisfies the equation for given conditions. In our case, when dealing with Laplace transforms, differential equations help express complex systems in terms of simple algebraic operations by transforming them into a different domain.

Improper Integral

An improper integral occurs when the limits of integration are infinite, or the integrand becomes infinite within the limits of integration. In the context of Laplace transforms, improper integrals appear as we evaluate the integral from zero to infinity. The challenge with improper integrals is ensuring they converge to a finite value. For instance, the Laplace transform of the function \( f(t) = e^{at} \) requires evaluating an improper integral. The convergence of this integral depends on the exponential term \( e^{(a-s)t} \), which decreases when \( s > a \), thereby ensuring the integral's convergence to a manageable value.

Convergence

Convergence is a vital concept when dealing with sequences, series, or integrals, as it determines if they tend towards a certain value. When we perform a Laplace transform, such as in the improper integral from zero to infinity, convergence ensures the result is finite and valid. Specifically, in the scenario of the Laplace transform \( \mathcal{L}\{e^{at}\} \), convergence occurs when \( s > a \). In simpler terms, convergence guarantees that as "t" tends towards infinity, the expression does not blow up but instead levels out to zero, thus making the Laplace transform meaningful and computable with a defined domain \( s > a \). This requirement assures the reliability of solutions derived through Laplace transforms for specific intervals.