Laplace Transforms and Differential Equations

Example 1 — Laplace Transform of a Derivative:

\[ \mathcal{L}\{f'(t)\} = sF(s) - f(0) \]

Example 2 — Solving a First-Order Differential Equation:

\[ \frac{dy}{dt} + 3y = e^{-2t}, \quad y(0) = 1 \]

Taking the Laplace transform of both sides:

\[ sY(s) - y(0) + 3Y(s) = \frac{1}{s + 2} \]

Solving for \( Y(s) \):

\[ Y(s) = \frac{1}{(s + 2)(s + 3)} + \frac{1}{s + 3} \]

Then use the inverse Laplace transform to find \( y(t) \).