Diff Eq final exam

Laplace and Variation in Parameters

y

Y

y’

sY - y(0)

y’’

s²Y - sy(0) - y’(0)

f(t) = 1

F(s) = 1 / s

f(t) = t

F(s) = 1 / s²

f(t) = tⁿ

F(s) = n! / sⁿ⁺¹

f(t) = e^at

F(s) = 1 / s - a

f(t) = sin(at)

F(s) = a / s² + a²

f(t) = cos(at)

F(s) = s / s² + a²

f(t) = e^at cos(bt)

F(s) = s - a / (s - a)² + b²

f(t) = e^at sin(bt)

F(s) = b / (s - a)² + b²

f(t) = tⁿ e^at

F(s) = n! / (s - a)ⁿ⁺¹

g(t) =

(first rule) + u_c(t)(second rule - first rule)

Wronskian =

y₁ y’₂ - y’₁ y₂

λ_1, λ₂ < 0

stable node and asymptotically stable

λ_1, λ₂ > 0

unstbale node and unstable

λ₁ < 0 < λ₂

saddle point and unstable

𝛼 = 0, β does not equal 0, λ = 𝛼 ^+_— βi

center and stable but NOT asymptotically

𝛼 > 0, β does not equal 0, λ = 𝛼 + βi

source and unstable

𝛼 < 0, β does not equal 0, λ = 𝛼 + βi

sink and asymptoticlaly stable

λ < 0 (defective)

stable improper node and symptotically stable

λ > 0 (defective)

unstable improper node and unstable