Questions: The Laplace transform of a continuous function over the interval [0, ∞) is defined by F(s)=∫ from 0 to ∞ e^(-sx) f(x) dx. This definition is used to solve some important initial-value problems in differential equations. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of the following function: f(x)=6x

The Laplace transform of a continuous function over the interval [0, ∞) is defined by F(s)=∫ from 0 to ∞ e^(-sx) f(x) dx. This definition is used to solve some important initial-value problems in differential equations. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of the following function:
f(x)=6x

Transcript text: The Laplace transform of a continuous function over the interval $[0, \infty)$ is defined by $F(s)=\int_{0}^{\infty} e^{-s x} f(x) d x$. This definition is used to solve some important initial-value problems in differential equations. The domain of $F$ is the set of all real numbers $s$ such that the improper integral converges. Find the Laplace transform $F$ of the following function: \[ f(x)=6 x \]

Solution

Solution Steps

Step 1: Define the Function

We start with the function $ f(x) = 6x $.

Step 2: Set Up the Laplace Transform

The Laplace transform $ F(s) $ is defined as: \[ F(s) = \int_{0}^{\infty} e^{-sx} f(x) \, dx \] Substituting $ f(x) $ into the equation, we have: \[ F(s) = \int_{0}^{\infty} e^{-sx} (6x) \, dx \]

Step 3: Evaluate the Integral

To evaluate the integral, we recognize that it can be computed using integration by parts. The result of the integral is: \[ F(s) = \frac{6}{s^2} \quad \text{for } |s| < \frac{\pi}{2} \]