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 \(F\).\(f(x)=\cos (2 x)\)

Short answer

The Laplace transform of \( \cos(2t) \) is \( \frac{s}{s^2 + 4} \) for \( s > 0 \).

Step-by-step solution

Write the Laplace Transform Formula

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 formula to \( f(t) = \cos(2t) \).

Substitute the Function into the Formula

Replace \( f(t) \) with \( \cos(2t) \) in the Laplace transform formula: \[ \mathcal{L}\{\cos(2t)\} = \int_{0}^{\infty} e^{-st} \cos(2t)\, dt. \]

Use the Known Laplace Transform

For \( \cos(at) \), the Laplace transform is \( \frac{s}{s^2 + a^2} \). Here, \( a = 2 \), so the transform is \[ \mathcal{L}\{\cos(2t)\} = \frac{s}{s^2 + 4}. \]

Determine the Domain of the Laplace Transform

The integral \[ \int_{0}^{\infty} e^{-st} \cos(2t)\, dt \] converges for \( s > 0 \). Therefore, the domain of \( F(s) = \frac{s}{s^2 + 4} \) is \( s > 0 \).

Key concepts

Initial-Value Problems

Initial-value problems are types of problems in differential equations where you know the values of the function and its derivatives at a specific starting point. This allows you to find a unique solution to the differential equation. These problems give a complete picture of how a system behaves over time. They are crucial in real-world applications, like determining how an electrical circuit responds to an initial surge of current, or how the temperature in a room changes when a heater is first turned on.

In initial-value problems, once you have the initial conditions, combined with the differential equation, you can use tools like the Laplace transform to find solutions. The Laplace transform turns a differential equation, which might be hard to solve directly, into an algebraic equation, which is usually easier to handle. This simplifies finding the specific function that satisfies both the differential equation and the given initial conditions.

Differential Equations

Differential equations involve equations with derivatives, indicating how a function changes. They play an essential role in fields like physics, engineering, and economics because they model various dynamic systems. The central feature of a differential equation is that it links a function to its derivatives.

For instance, consider an equation that displays how a population grows over time based on its current size and rate of growth. This scenario can be represented by a differential equation. Solving these equations can sometimes be tricky, but they provide insightful solutions about the system's behavior, enabling predictions of future states under certain conditions.

Typically, solving such equations directly requires advanced techniques. Hence, transformation methods like the Laplace Transform become invaluable tools in simplifying and finding solutions to these problems.

Improper Integral

Improper integrals extend the concept of integrals to cases where the interval of integration is infinite or the integrand has infinite discontinuities. For the Laplace Transform, we deal with improper integrals because we integrate from zero to infinity.

To work with improper integrals, we often use a limit process. This involves finding the integral over a finite interval and then taking the limit as the interval extends to infinity. The convergence or divergence of these integrals depends on the behavior of the function involved. For example, the function must decrease quickly enough as its argument increases toward infinity.

In the context of Laplace transforms, the convergence of these improper integrals determines the domain of the transformed function. If the integral converges, the Laplace Transform exists, which then allows us to further solve differential equations.

Laplace Transform Formula

The Laplace Transform formula is a powerful tool used to transform a function into a simpler form, often turning a differential equation into an algebraic equation. The transform is given by:\[ \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t)\, dt \] This formula involves an integral from 0 to infinity of the original function multiplied by an exponential decay term.

The real strength of the Laplace transform is in its ability to simplify the process of solving differential equations. By converting the functions and operations involved, it can make complex, time-dependent problems much easier to manage. The formula utilizes the convergence of the improper integral to establish the domain of the resulting transformed function. This convergence is dictated by the mathematical behavior of the functions involved.

• Converts differential equations to algebraic ones
• Helps find solutions for initial-value problems
• Provides insight into system behavior over time