Laplace transforms, step functions, systems of equations, phase portraits, stability and power series solutions.
42 cards · basic cards · AI-written, checked twice. Edit anything.
- Define the Laplace transform L{f(t)}.
Show answer
L{f(t)} = ∫[0 to ∞] e^(-st)f(t) dt, where s is a complex parameter.- State the linearity property of the Laplace transform.
Show answer
L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)} for constants a, b.- State the frequency shift (s-shift) property.
Show answer
L{e^(at)f(t)} = F(s-a) where F(s) = L{f(t)}.- What is the derivative property of the Laplace transform?
Show answer
L{f'(t)} = sF(s) - f(0), or more generally L{f^(n)(t)} = s^n F(s) - s^(n-1)f(0) - ... - f^(n-1)(0).- State the integration property of the Laplace transform.
Show answer
L{∫[0 to t] f(τ) dτ} = F(s)/s.- What is L{t^n}?
Show answer
L{t^n} = n! / s^(n+1) for n = 0, 1, 2, ...- What is L{e^(at)}?
Show answer
L{e^(at)} = 1/(s-a) for s > a.- What is L{sin(bt)}?
Show answer
L{sin(bt)} = b/(s^2 + b^2) for s > 0.- What is L{cos(bt)}?
Show answer
L{cos(bt)} = s/(s^2 + b^2) for s > 0.- Define the inverse Laplace transform.
Show answer
The inverse Laplace transform L^(-1){F(s)} is the function f(t) such that L{f(t)} = F(s).- Describe the partial fraction decomposition method for finding inverse Laplace transforms.
Show answer
Decompose F(s) into simpler fractions using partial fractions, then apply L^(-1) to each term using a table of known transforms.- Define the Heaviside (unit step) function u_c(t).
Show answer
u_c(t) = 0 for t < c, and u_c(t) = 1 for t >= c.- What is L{u_c(t)}?
Show answer
L{u_c(t)} = e^(-cs)/s for s > 0.- Define the convolution of two functions f and g.
Show answer
(f * g)(t) = ∫[0 to t] f(τ)g(t-τ) dτ.- State the convolution theorem.
Show answer
L{(f * g)(t)} = F(s)G(s) where F = L{f}, G = L{g}.
FlashKeepers