Problem 82
Question
Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=t $$
Step-by-Step Solution
Verified Answer
The Laplace transform of the function \(f(t) = t\) is \(F(s) = 1/s^{2}\) for \(s > 0\).
1Step 1: Substitute the function into the formula
Replace \(f(t)\) in the integral with the function \(t\). The integral to solve is given as \(F(s) = \int_{0}^{\infty} e^{-st}t dt\)
2Step 2: Solve the integral
Next, apply integration by parts to solve the integral. Let \(u = t\) and \(dv = e^{-st}dt\). Apply the formula for integration by parts: \(\int udv = uv - \int v du\). After calculating and evaluating the integral from \(0\) to \(\infty\), find the resulting value of \(F(s)\).
3Step 3: Substitute the limits
Calculation of the limits for the improper integral gives a finite value, \(F(s)\) if \(s > 0\). You must apply these limits to obtain the final result
Key Concepts
Differential EquationsImproper IntegralIntegration by Parts
Differential Equations
Differential equations are mathematical equations that involve functions and their derivatives. These equations describe how a certain quantity changes with respect to a variable, usually time. Differential equations are powerful tools in mathematics and engineering to model real-world problems like population growth, heat conduction, and motion dynamics.
Thus, applying a Laplace Transform to a differential equation allows us to work in a transformed space, making the solution process more straightforward and efficient.
- They can be classified into ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of one variable and their derivatives, while PDEs involve functions of multiple variables.
- Laplace Transforms are particularly useful in solving linear ODEs with constant coefficients, as they simplify the process by converting differential equations into algebraic ones.
Thus, applying a Laplace Transform to a differential equation allows us to work in a transformed space, making the solution process more straightforward and efficient.
Improper Integral
An improper integral is a type of definite integral where either the interval of integration is infinite, or the function being integrated has an undefined point within the interval. In many cases, these integrals appear in real-world applications, requiring sophisticated techniques for evaluation.
Understanding and calculating improper integrals is critical for working with Laplace Transforms, as they form the basis for transforming functions from the time domain to the frequency domain.
- The Laplace Transform itself is an example of an improper integral, as it spans an infinite domain from zero to infinity.
- To evaluate an improper integral, a limit is taken to consider the behavior of the integral as you approach the point of discontinuity or the infinite limit.
Understanding and calculating improper integrals is critical for working with Laplace Transforms, as they form the basis for transforming functions from the time domain to the frequency domain.
Integration by Parts
Integration by parts is a technique used to calculate integrals where the standard methods of integration are not applicable. It is particularly useful when integrating the product of two functions. The formula is: \[ \int u \, dv = uv - \int v \, du \].
It's vital to choose \(u\) and \(dv\) appropriately, as the complexity of the integral is determined by these selections. Mastering integration by parts can simplify many challenging integral problems, especially those encountered in differential equations.
- This technique requires selecting parts of the integrand as \(u\) and \(dv\), such that the remaining integration becomes easier.
- In the context of finding Laplace Transforms, integration by parts allows us to break down complex integrals like \( \int_{0}^{fty} e^{-st} t \, dt\) into simpler components that can be solved systematically.
It's vital to choose \(u\) and \(dv\) appropriately, as the complexity of the integral is determined by these selections. Mastering integration by parts can simplify many challenging integral problems, especially those encountered in differential equations.
Other exercises in this chapter
Problem 81
Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty}
View solution Problem 81
Consider the limit \(\lim _{x \rightarrow 0^{+}}(-x \ln x)\) (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate t
View solution Problem 82
Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=\infty,\) then \(\lim _{x \rightarrow a} f(x)^{g(x)}=0\).
View solution Problem 83
Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty}
View solution