Dr Denise Race
Eastfield
MATH 2420 FINAL EXAM FORMULAS
1. Laplace Transform
(a) Definition
∞
ℒ{𝑓(𝑡)} = ∫ 𝑒 −𝑠𝑡 𝑓(𝑡)𝑑𝑡
0
(b) Laplace transforms of some common functions
1
ℒ{1} =
𝑠
𝑛!
ℒ{𝑡 𝑛 } = 𝑛+1
𝑠
𝑘
ℒ{sin(𝑘𝑡)} = 2
𝑠 + 𝑘2
𝑠
ℒ{cos(𝑘𝑡)} = 2
𝑠 + 𝑘2
1
ℒ{𝑒 𝑎𝑡 } =
𝑠−𝑎
𝑎𝑡
ℒ{𝑒 𝑓(𝑡)} = 𝐹(𝑠 − 𝑎) = ℒ{𝑓(𝑡)}|𝑠→𝑠−𝑎
𝑑𝑛
𝑛
𝑛
[𝐹(𝑠)]
ℒ{𝑡 𝑓(𝑡)} = (−1)
𝑑𝑠 𝑛
(c) Derivatives
ℒ{𝑓 ′′ (𝑡)} = 𝑠 2 𝐹(𝑠) − 𝑠𝑓(0) − 𝑓′(0)
ℒ{𝑓 ′ (𝑡)} = 𝑠𝐹(𝑠) − 𝑓(0)
2. Indicial Equation
If 𝑥0 is a regular singular point of 𝑦 ′′ + 𝑝𝑦 ′ + 𝑞𝑦 = 0, then the indicial equation for this point
is
𝑟(𝑟 − 1) + 𝑝0 𝑟 + 𝑞0 = 0,
where
𝑝0 = lim (𝑥 − 𝑥0 )𝑝(𝑥) and 𝑞0 = lim (𝑥 − 𝑥0 )2 𝑞(𝑥)
𝑥→𝑥0
𝑥→𝑥0
3. Variation of Parameters
A general solution to 𝒙′ (𝑡) = 𝑨(𝑡)𝒙(𝑡) is given by 𝒙(𝑡) = 𝑿(𝑡)𝑪, where 𝑿(𝑡) is the
fundamental matrix for the homogeneous system.
A particular solution to the nonhomogeneous system 𝒙′ (𝑡) = 𝑨(𝑡)𝒙(𝑡) + 𝒇(𝑡) is of the form
𝑣1 (𝑡)
𝒙𝑝 (𝑡) = 𝑿(𝑡)𝒗(𝑡), where 𝒗(𝑡) = [ ⋮ ] can be found by
𝑣𝑛 (𝑡)
𝒗(𝑡) = ∫ 𝑿−1 (𝑡)𝒇(𝑡)𝑑𝑡
Revised 1/1/2020