Understanding Laplace Transform and Its Applications

Laplace Transform - Purpose

A method used to convert differential equations into algebraic equations, making them easier to solve.

Unique Capabilities of Laplace Transform

It can handle (1) discontinuous and (2) impulsive forces

Partial Fractions - Subcases

(1) Distinct linear factors, (2) repeated linear factors, and (3) irreducible quadratic factors.

A/s,

Bs+c/s^2,

Ds+E/s^2+4

Before Using Partial Fractions

Ensure the degree of the numerator is less than the degree of the denominator

Order of Operations - Partial Fractions vs Completing the Square

Do partial fractions first or you will pay

Use of Step Function u(t−a)

Models functions that 'turn on' at a certain time t=a

Real-World Use of Step Function

Turning on a heater at t=5 minutes—modeled as u(t−5).

Use of Delta Function δ(t−a)

Represents an instantaneous impulse or spike at time t=a, such as a sudden force or voltage jolt.

Real-World Use of Delta Function

A hammer striking a nail at exactly t=2 seconds—modeled as δ(t−2).