8 - Laplace Transforms & Applications

Welcome class! Today, we will explore an important property of Laplace transforms called division by t. This property connects time domain operations with something we can work with in the s-domain.

Good question! The division by t rule states that if we know the Laplace transform of a function, we can find the transform of its division by t by integrating the s-domain function. For instance, if \( F(s) = \mathcal{L}\{f(t)\} \), then \( \mathcal{L}\left\{ \frac{f(t)}{t} \right\} = \int_{0}^{\infty} F(u) du \, t \).

Exactly! It's a key technique that simplifies complex problems. Remember, this approach is particularly useful if \( f(t) \) is piecewise continuous and of exponential order.

Certainly! This property is valuable in control systems for analyzing dynamic systems and in electrical engineering to assess decaying signals during transient responses.

Let's summarize: the division by t rule in Laplace transforms helps translate time domain operations to manageable algebraic forms in the s-domain, facilitating various engineering applications.

Now that we've introduced the division by t property, let’s delve into its mathematical formulation.

The expression is quite simple. Recall that for a function \( F(s) \), the Laplace of \( \frac{f(t)}{t} \) is given by \( \mathcal{L}\left\{ \frac{f(t)}{t} \right\} = \int_{0}^{\infty} F(u) du \, t \).

Great inquiry! This switch allows us to leverage known results of \( F(s) \) for more complex derivations. It uses the inverse Laplace transform to connect the time domain and s-domain seamlessly.

Of course. It's crucial that \( f(t) \) is piecewise continuous and exhibits exponential order; otherwise, the Laplace transform may not exist.

Absolutely! Using this rule properly depends on our knowledge of the function’s characteristics. Let's wrap up this session: Division by t links time domain operations to integral formulations in s-domain, crucial for solving differential equations and analyzing systems.

Now, let’s look at some examples to see how the division by t rule is applied.

Certainly! For \( \sin(at) \), we know that its Laplace transform is \( F(s) = \frac{a}{s^2 + a^2} \). By applying the division by t property, we get:\n \( \mathcal{L}\left\{ \frac{\sin(at)}{t} \right\} = \int_{0}^{\infty} \frac{a}{u^2 + a^2} du \).

Good question! This integral can be solved as \( \frac{a}{s} \left[ \tan^{-1}(\frac{u}{a}) \right]_{0}^{\infty} = \frac{\pi a}{2s} \).

For \( 1 - \cos(at) \), the Laplace transform is given as \( \frac{a^2}{s(s^2 + a^2)} \). Using the division by t rule gives us another integral to consider.

Yes! But many times, we would look up the result in Laplace tables. In the end, these examples showcase how effective this property is for finding Laplace transforms.

Let’s recap: we discussed practical applications of the division by t rule using sine and cosine functions, illustrating its usefulness in transforming and manipulating signals and equations.