The Laplace Transform and its Applications

Laplace transform

The Laplace transform of a function f(t) is calculated as

\[ F(s) = \int_{0}^{\infty} f(t)e^{-st}\,dt \]

Notation:

\[ F(s) = \mathcal{L}\{f(t)\} \]

Inverse transform:

\[ f(t) = \mathcal{L}^{-1}\{F(s)\} \]

Example: Find the Laplace transform of f(t) = a

Example: Find the Laplace transform of f(t) = 2 + 5t

Example: Find the Laplace transform of f(t) = t

Inverse Laplace Transform Problem

Find the inverse Laplace transform of F(s) = (3s - 4) / (s² + 9)

Solving linear ODEs using the Laplace transform

Problem 1: Solve the ODE 2 0 with the initial condition df/dt + 2f = 0 with the initial condition f(0) = 5

Problem 2: Solve the ODE y" + 4y = 0 with the initial conditions y(0) = 0 y'(0) = 0

Problem 3: Solve the ODE f" + 4f = 4cos(3t) using the initial conditions f(0) = 0 and f'(0) = 1

Laplace transform: Test yourself

Using the table below:

Test your knowledge as follows

Useful Video tutorials on Partial Fraction Decomposition: