Question

The first shift theorem states that if \(\mathcal{L}(f(t)\\}=F(s)\), then $$ \mathcal{L}\left\\{\mathrm{e}^{-a t} f(t)\right\\}=F(s+a) $$ where \(a\) is a constant. (a) From the definition of the Laplace transform show that $$ \mathcal{L}\left\\{\mathrm{e}^{-a t} f(t)\right\\}=\int_{0}^{\infty} \mathrm{e}^{-(s+a) t} f(t) \mathrm{d} t $$ and hence prove the first shift theorem. (b) Use Table \(1.1\) in Block 1 and the first shift theorem to find \(\mathcal{L}\left\\{u(t-3) \mathrm{e}^{-7 t}\right\\}\) where \(u(t)\) is the unit step function.

Short answer

Question: Prove the first shift theorem for Laplace Transform and use it to find the Laplace Transform of the given function: 𝑢(𝑡−3)𝑒^(−7𝑡). Answer: The first shift theorem for the Laplace Transform states that if the Laplace transform of a function 𝑓(t) is F(s), then the Laplace transform of 𝑒^(−𝑎t)𝑓(t) is F(s+a). Using the proof outlined in the solution, we can find the Laplace Transform of the given function 𝑢(𝑡−3)𝑒^(−7𝑡) as follows: $$ \mathcal{L}\left\\{u(t-3) \mathrm{e}^{-7 t}\right\\} = \frac{1}{s + 7} $$

Step-by-step solution

Part (a) - Proving the first shift theorem

Step 1: Use the definition of the Laplace transform To prove the first shift theorem, let's write down the definition of the Laplace transform of 𝑒^(−𝑎𝑡)𝑓(𝑡): $$ \mathcal{L}\left\\{\mathrm{e}^{-a t} f(t)\right\\} = \int_{0}^{\infty} \mathrm{e}^{-s t} \left( \mathrm{e}^{-a t} f(t) \right) \mathrm{d} t $$ Step 2: Integrate using the substitution method Now, let's change the integral by combining the exponential terms: $$ \mathcal{L}\left\\{\mathrm{e}^{-a t} f(t)\right\\} = \int_{0}^{\infty} \mathrm{e}^{-(s + a) t} f(t) \mathrm{d} t $$ Step 3: Apply the first shift theorem The above integral is the Laplace transform of the function 𝑓(𝑡) with variable (𝑠+𝑎), so we can rewrite it as: $$ \mathcal{L}\left\\{\mathrm{e}^{-a t} f(t)\right\\} = F(s + a) $$ This proves the first shift theorem.

Part (b) - Finding the Laplace Transform using the first shift theorem

Step 1: Rewrite the given function using unit step function The given function is: $$ u(t - 3) \mathrm{e}^{-7 t} $$ Step 2: Apply the first shift theorem According to the first shift theorem: $$ \mathcal{L}\left\\{\mathrm{e}^{-7 t} u(t - 3)\right\\} = U(s + 7) $$ where 𝑈(𝑠) is the Laplace transform of the unit step function. Step 3: Find the Laplace transform of the unit step function using Table 1.1 From Table 1.1, we know that the Laplace transform of the unit step function is: $$ \mathcal{L}\{u(t)\} = \frac{1}{s} $$ Step 4: Apply the first shift theorem to find the Laplace Transform of 𝑢(𝑡−3)𝑒^(−7𝑡) Replacing 𝑈(𝑠) with its Laplace Transform, we get: $$ \mathcal{L}\left\\{u(t-3) \mathrm{e}^{-7 t}\right\\} = \frac{1}{s + 7} $$ Thus, the Laplace Transform of the given function is: $$ \mathcal{L}\left\\{u(t-3) \mathrm{e}^{-7 t}\right\\} = \frac{1}{s + 7} $$

Key concepts

First Shift Theorem

The First Shift Theorem is a valuable tool in the study of Laplace transforms, especially when dealing with exponential functions. This theorem relates the Laplace transform of an exponentially shifted function to its original Laplace transform. The concept is simple: if you know the Laplace transform of a function, the First Shift Theorem allows you to easily find the transform of the same function multiplied by an exponential decay or growth term.

Imagine you have function, say, f(t), and you want to determine how its behavior changes when it is multiplied by an exponential term, e^(-at). According to the theorem, if the Laplace transform of f(t) is F(s), then the transform of e^(-at)f(t) simply shifts the variable s by a, becoming F(s + a). This principle makes analyzing systems affected by exponential terms straightforward, as you can start with the original function's transform without the need for additional complex integrations.

Unit Step Function

The unit step function, commonly represented as u(t), is a fundamental concept in signal processing and control theory. It is defined to be zero for all negative values and one for positive values, effectively 'turning on' a signal at a specified time. The function acts as a switch, where the 'on' value can represent the start of a process or the application of a force.

When dealing with Laplace transforms, the unit step function is used to model sudden changes or shifts in a system's behavior. For instance, if you want to apply a force to a system starting at time t = 3, you would use the unit step function u(t - 3). The Laplace transform of this delayed step can be found using a table of transforms, but in essence, it helps isolate events in time within complex systems, allowing for a modular approach to solving time-dependent differential equations.

Integration by Substitution

Integration by substitution, also known as u-substitution, is a mathematical technique used to simplify complex integrals. It's analogous to the change of variables in algebra, enabling us to turn a challenging integral into an easier one by switching variables. This method is particularly useful when dealing with integrals involving compositions of functions, such as products of polynomials and exponentials, or nested functions.

When proving the First Shift Theorem, integration by substitution makes the process manageable by combining the exponential terms of the integrand into a single exponent. This effectively transforms the original integral, which may be difficult to evaluate as is, into a form that is straightforward to integrate. It's like finding a familiar path through a complicated forest. For students, mastering this technique can be incredibly empowering, turning the maze of integration into a structured, navigable system.

Exponential Functions

Exponential functions are vital in many fields, including mathematics, physics, engineering, and finance, because they describe growth and decay processes. An exponential function has the form f(t) = e^(kt), where e is the base of natural logarithms, and k is a constant that determines the growth (if positive) or decay (if negative) rate.

When we apply the Laplace transform to exponential functions, it allows us to move from the time domain to the frequency domain. This transformation is significant in solving differential equations that describe real-world systems such as electronic circuits, mechanical vibrations, and population models. Understanding exponential functions and their behavior through the lens of Laplace transforms is therefore crucial for students who wish to model and solve time-dependent phenomena efficiently.