Laplace transform: F(s) = ∫₀^∞ f(t)e⁻ˢᵗdt. Converts ODEs to algebraic equations. L{eᵃᵗ} = 1/(s−a). L{tⁿ} = n!/sⁿ⁺¹. L{sin(ωt)} = ω/(s²+ω²). Poles = natural frequencies. Transfer function H(s) = output/input in s-domain.
How to Use
Select a function:
- f(t): Time-domain function
- F(s): s-domain transform
- Analysis: Poles, zeros
ODE Solving
L{y'} = sY(s)−y(0). L{y''} = s²Y(s)−sy(0)−y'(0). Transform the ODE, solve for Y(s) algebraically, then inverse transform. Initial conditions are built in!
Stability
System stable ⟺ all poles have Re(s) < 0 (left half plane). Poles on imaginary axis = marginally stable. Right half plane = unstable. Routh-Hurwitz criterion checks this without finding poles.
Step-by-Step Instructions
- 1Select a function.
- 2View F(s).
- 3Find poles/zeros.
- 4Check properties.
- 5Use for ODEs.