Diff Eq Final Exam

sin(2t) =

2 sin(t) cos(t)

used to cancel terms in the denominator w/ tan(t) or sec(t)

cos(2t) =

cos²(t) - sin²(t)

2 cos²(t) - 1

1 - 2 sin²(t)

used to rewrite terms before integrating

sin²(t) =

1 - cos(2t) / 2

used when integral has a squared term in it

cos²(t) =

1 + cos(2t) / 2

used when integral has asquared term in it

tan(t) =

sin(t) / cos(t)

sec(t) =

1 / cos(t)

csc(t) =

1 / sin(t)

cot(t) =

cos(t) / sin(t)

cos²(t) + sin²(t) =

1

1 + tan²(t) =

sec²(t)

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