Steps to Apply Laplace Transforms for Periodic Inputs in Differential Equations

📚 Introduction to Laplace Transforms for Periodic Inputs

Laplace Transforms are a powerful tool for solving linear differential equations, especially those with periodic forcing functions. The key is understanding how to represent these periodic functions in the s-domain. Here's a comprehensive guide:

📜 History and Background

The Laplace Transform, named after Pierre-Simon Laplace, evolved from his work on probability theory. It provides a method for solving differential equations by transforming them into algebraic equations. This approach simplifies many problems, particularly those involving discontinuous or periodic inputs.

🔑 Key Principles

• 🔍 Definition: The Laplace Transform of a function $f(t)$, denoted by $F(s)$, is defined as: $F(s) = \int_0^\infty e^{-st}f(t) dt$
• ⏱️ Time Period (T): For a periodic function $f(t)$ with period $T$, $f(t+T) = f(t)$ for all $t$.
• 🔄 Laplace Transform of a Periodic Function: The Laplace Transform of a periodic function $f(t)$ with period $T$ is given by: $F(s) = \frac{1}{1 - e^{-sT}} \int_0^T e^{-st} f(t) dt$
• ➕ Linearity: The Laplace transform is a linear operator: $\mathcal{L}[af(t) + bg(t)] = a\mathcal{L}[f(t)] + b\mathcal{L}[g(t)]$
• 📉 Shifting Theorem: If $\mathcal{L}[f(t)] = F(s)$, then $\mathcal{L}[f(t-a)u(t-a)] = e^{-as}F(s)$, where $u(t)$ is the Heaviside step function.

🪜 Steps to Apply Laplace Transforms for Periodic Inputs

• 📝 Step 1: Identify the Periodic Function: Determine the function $f(t)$ that represents the periodic input. Also, find its period $T$.
• ➕ Step 2: Define the Function Over One Period: Express $f(t)$ for $0 \le t < T$. This is crucial for calculating the integral in the Laplace Transform formula.
• 🧮 Step 3: Calculate the Integral: Compute the integral $\int_0^T e^{-st} f(t) dt$. This step often involves integration by parts or using known Laplace Transforms of simpler functions.
• ➗ Step 4: Apply the Formula: Use the formula for the Laplace Transform of a periodic function: $F(s) = \frac{1}{1 - e^{-sT}} \int_0^T e^{-st} f(t) dt$. Substitute the result from Step 3 into this formula.
• ✍️ Step 5: Solve the Differential Equation: Substitute $F(s)$ into the transformed differential equation. Solve for the Laplace Transform of the output, $Y(s)$.
• 🔀 Step 6: Inverse Laplace Transform: Find the inverse Laplace Transform of $Y(s)$ to obtain the solution $y(t)$ in the time domain. This may require partial fraction decomposition or using Laplace Transform tables.

💡 Real-World Examples

Example 1: Square Wave Input

Consider a differential equation with a square wave input $f(t)$ of amplitude $A$ and period $T$, defined as:

$f(t) = A$ for $0 \le t < T/2$

$f(t) = -A$ for $T/2 \le t < T$

• 1️⃣ Step 1: $f(t)$ is a square wave with period $T$.
• 2️⃣ Step 2: Define $f(t)$ as above.
• 3️⃣ Step 3: Calculate $\int_0^T e^{-st} f(t) dt = \int_0^{T/2} Ae^{-st} dt + \int_{T/2}^T -Ae^{-st} dt = \frac{A}{s}(1 - 2e^{-sT/2} + e^{-sT})$.
• 4️⃣ Step 4: $F(s) = \frac{1}{1 - e^{-sT}} \cdot \frac{A}{s}(1 - 2e^{-sT/2} + e^{-sT}) = \frac{A}{s} \cdot \frac{1 - e^{-sT/2}}{1 + e^{-sT/2}} = \frac{A}{s} \tanh(\frac{sT}{4})$.

Example 2: Sawtooth Wave Input

Consider a sawtooth wave defined as $f(t) = At/T$ for $0 \le t < T$.

• 1️⃣ Step 1: $f(t)$ is a sawtooth wave with period $T$.
• 2️⃣ Step 2: Define $f(t) = At/T$ for $0 \le t < T$.
• 3️⃣ Step 3: Calculate $\int_0^T e^{-st} \frac{At}{T} dt = \frac{A}{T} \int_0^T te^{-st} dt = \frac{A}{T} [-\frac{t}{s}e^{-st} - \frac{1}{s^2}e^{-st}]_0^T = \frac{A}{T} [-\frac{T}{s}e^{-sT} - \frac{1}{s^2}e^{-sT} + \frac{1}{s^2}]$.
• 4️⃣ Step 4: $F(s) = \frac{1}{1 - e^{-sT}} \cdot \frac{A}{T} [-\frac{T}{s}e^{-sT} - \frac{1}{s^2}e^{-sT} + \frac{1}{s^2}] = \frac{A}{T} \frac{1 - e^{-sT} - sTe^{-sT}}{s^2(1 - e^{-sT})}$.

📝 Conclusion

Applying Laplace Transforms to differential equations with periodic inputs involves understanding the periodic nature of the input and using the appropriate Laplace Transform formula. This method simplifies the solution process by converting the differential equation into an algebraic one, making it easier to solve. By following these steps, you can effectively analyze and solve a wide range of engineering and physics problems involving periodic forcing functions.