How To Use Laplace Inverse? Stepbystep Solutions

The Laplace transform is a powerful tool for solving differential equations, and its inverse is equally important for transforming the solution back to the original domain. Here’s a step-by-step guide on how to use the Laplace inverse:

Understanding the Laplace Inverse

The Laplace inverse, denoted as L^(-1), is an operator that takes a function F(s) in the s-domain and transforms it back to the time domain, resulting in a function f(t). The Laplace inverse is defined as:

L^(-1) {F(s)} = f(t)

Step-by-Step Solution

To apply the Laplace inverse, follow these steps:

1. Identify the function F(s): Start with the function F(s) in the s-domain, which is the result of applying the Laplace transform to a function f(t).
2. Check the Laplace transform tables: Look up the Laplace transform tables to find the corresponding inverse transform. The tables provide a list of common functions and their Laplace transforms.
3. Use the linearity property: If F(s) is a linear combination of functions, use the linearity property of the Laplace inverse to break down the function into simpler components.
4. Apply the inverse Laplace transform formula: Use the formula for the inverse Laplace transform, which is given by:

L^(-1) {F(s)} = (1/2πi) * ∫_{σ-i∞}^{σ+i∞} F(s) * e^(st) ds

where σ is a real number and i is the imaginary unit. 5. Evaluate the integral: Evaluate the integral using techniques such as partial fractions, contour integration, or other methods. 6. Simplify the result: Simplify the resulting expression to obtain the final answer in the time domain.

Example 1: Finding the Inverse Laplace Transform of 1/(s+1)

Suppose we want to find the inverse Laplace transform of F(s) = 1/(s+1).

1. Check the Laplace transform tables: The table shows that L^(-1) {1/(s+1)} = e^(-t).
2. No need to apply linearity property or inverse Laplace transform formula, as the result is directly available from the tables.

Therefore, the inverse Laplace transform of 1/(s+1) is e^(-t).

Example 2: Finding the Inverse Laplace Transform of (2s+1)/(s^2+4)

Suppose we want to find the inverse Laplace transform of F(s) = (2s+1)/(s^2+4).

1. Check the Laplace transform tables: The table does not provide a direct match, so we need to apply the linearity property and inverse Laplace transform formula.
2. Apply linearity property: Break down the function into simpler components: F(s) = 2s/(s^2+4) + 1/(s^2+4).
3. Apply inverse Laplace transform formula: Use the formula to evaluate the inverse Laplace transform of each component.

After evaluating the integrals and simplifying the result, we obtain:

L^(-1) {(2s+1)/(s^2+4)} = 2cos(2t) + (¹⁄₂)sin(2t)

Common Laplace Inverse Formulas

Here are some common Laplace inverse formulas:

• L^(-1) {1/s} = 1
• L^(-1) {1/s^2} = t
• L^(-1) {1/s^3} = (¹⁄₂)t^2
• L^(-1) {e^(-as)}/s = (1 - e^(-at)) / t
• L^(-1) {sin(at)}/s = (1 - cos(at)) / at

Conclusion

The Laplace inverse is a powerful tool for transforming functions from the s-domain back to the time domain. By following the step-by-step solution and using the Laplace transform tables, linearity property, and inverse Laplace transform formula, you can easily find the inverse Laplace transform of a wide range of functions. Remember to simplify the result and check the answer using common Laplace inverse formulas.

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