Engineering Mathematics I
Laplace Transforms
Module IV: Laplace Transforms. This unit introduces the Laplace transform, inverse transform, properties and its use in solving ordinary differential equations. Short examples are included for quick learning.
Definition of Laplace Transform
- Laplace transform of f(t):
F(s) = ∫₀^∞ e^(-st) f(t) dt. - Exists if f(t) is of exponential order.
- Example:
L{1} = 1/s , L{e^(at)} = 1/(s-a).
Diagram: laplace-transform-idea.png
Properties of Laplace Transform
- Linearity:
L{af + bg} = aF + bG. - First Shifting:
L{e^(at) f(t)} = F(s-a). - Differentiation in time:
L{f'(t)} = sF(s) - f(0). - Integration in time:
L{∫ f(t) dt} = F(s)/s. - Example:
L{t} = 1/s^2.
Inverse Laplace Transform
- Obtained using partial fractions or standard transform pairs.
- Example:
L⁻¹{1/(s^2 + 4)} = (1/2) sin(2t).
Solving ODEs Using Laplace Transform
- Steps:
- Apply Laplace transform on both sides.
- Use initial conditions.
- Solve algebraic equation for F(s).
- Take inverse Laplace transform.
- Example:
y' + 2y = 4 , y(0)=1givesY(s) = (1 + 4/s)/(s+2).
Diagram: laplace-solve-ode.png