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 \(F\).\(f(x)=1\)

Short answer

\( F(s) = \frac{1}{s} \) with domain \( s > 0 \).

Step-by-step solution

- Write the Definition of the Laplace Transform

The Laplace transform of a function \( f(t) \) is given by \[ F(s) = \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt \]We need to apply this definition to the function \( f(t) = 1 \).

- Substitute the Function into the Integral

Substitute \( f(t) = 1 \) into the Laplace transform equation:\[ F(s) = \int_{0}^{\infty} e^{-st} \cdot 1 \, dt = \int_{0}^{\infty} e^{-st} \, dt \]

- Solve the Integral

The integral \( \int_{0}^{\infty} e^{-st} \, dt \) is evaluated as follows: \[ = \left[ \frac{-e^{-st}}{s} \right]_{0}^{\infty} \] \[ = \left( \lim_{b \to \infty} \frac{-e^{-sb}}{s} \right) - \left( \frac{-e^{-s \times 0}}{s} \right) \] \[ = \left( 0 \right) - \left( \frac{-1}{s} \right) = \frac{1}{s} \]This integration is valid for \( s > 0 \).

- Conclude the Domain

The domain of the Laplace transform \( F(s) \) is the set of all real numbers \( s \) for which the integral converges. Since the integral \( \int_{0}^{\infty} e^{-st} \, dt \) converges for \( s > 0 \), the domain of \( F(s) \) is \( s > 0 \).

Key concepts

Initial Value Problems

In the study of differential equations, initial value problems (IVPs) are a foundational concept. An initial value problem involves finding a function that satisfies a differential equation and meets specified conditions at a starting point, known as the initial conditions. For instance, suppose we have a differential equation in the form \( y'(t) = f(t, y) \) with an initial condition \( y(t_0) = y_0 \). Our task is to find the function \( y(t) \) that satisfies both the equation and the initial condition.

• The initial condition provides a specific value at the initial point, grounding the abstract nature of differential equations to practical scenarios.
• This ensures the uniqueness of the solution in many cases, meaning that for given initial conditions, there is typically one solution that fits.
  
• Applications of IVPs span from physics and engineering to economics and biology, such as predicting the motion of objects, population growth, or financial forecasting.

Differential Equations

Differential equations are equations that involve the rates of change of quantities and the quantities themselves. The concept is central in mathematics as it connects algebra with calculus and fosters our understanding of dynamic systems. Differential equations may be classified into ordinary differential equations (ODEs) and partial differential equations (PDEs).

• Ordinary differential equations (ODEs) involve functions of a single variable and their derivatives. For example, Newton's law of cooling, which describes how the temperature of an object changes over time.
• Partial differential equations (PDEs) involve functions of multiple variables and their partial derivatives, like the equation for heat conduction in a rod.
  
• Solutions to differential equations give us insights into the behavior of systems over time, predicting future states based on current conditions.

Understanding how to work with them, like finding Laplace transforms, allows for more straightforward solutions to these complex equations.

Improper Integral Convergence

Improper integrals are integrals with infinite limits or integrands that approach infinity within the integration range. Their convergence determines the existence of a numerical value. For Laplace transforms, improper integrals are crucial because the definition involves integrating functions over infinite intervals.

• An improper integral converges if its value approaches a finite limit as the interval of integration extends to infinity.
  
• For the Laplace transform, convergence typically requires that the real part of \( s \) (in the transform variable) is positive, ensuring that the function approaches zero as it tends to infinity.
• This aspect of convergence is essential to ensure that the Laplace transform of a function exists and can be used to solve problems involving differential equations.

By considering conditions for convergence, like \( s > 0 \) in our example, we confirm that the transform provides meaningful analytical solutions.