Represented by Dr. Shorouk Ossama (Laplace Transform) Represented by Dr. Shorouk Ossama

Definition We are going to study of transforming differential equations into algebraic equations. Laplace Transform L for the function f (t) is defined by:

Example: Find Laplace Transform For f (t): Solution a)The Laplace transform for f (t) = 1 given by: a)The Laplace transform for f (t) = t given by:

In general

Example: Find Laplace Transform For: 𝒆𝒂𝒕 , sin at , cos at, sinh at, cosh at Solution

Theorem (1): Linearity The Laplace Transform has these inherited integral properties: a) L{ f (t) + g (t) } = L{ f (t)} + L {g (t) } b) L{ c f (t) } = L c { f (t) } Example: Let f (t) = t (t – 1) – sin 2t + e3t , compute L{f (t)} Solution L {f (t)} = L { t2 – t – sin 2t + e3t } = L {t2} –L{t} –L{sin 2t} + L {e3t } = 2 𝑠 3 − 1 𝑠 2 − 2 𝑠 2 + 4 + 1 𝑠−3

Example: Find The Laplace Transform For The Following (a) f (t) = 3 t2 + sin 5t

𝟏 𝟐 L { 1 – cos 4t }

Theorem (2): The s- Differential Rule Let f (t) be of exponential order, then L{ t f (t) } = - 𝒅 𝒅𝒔 𝑳 𝒇 𝒕 L{ 𝒕 𝒏 f (t) } = (−𝟏) 𝒏 𝒅 𝒏 𝒅 𝒔 𝒏 𝑳 𝒇 𝒕 Example: Find The Following (a) L {t cos at } 𝑮𝒆𝒏𝒆𝒓𝒂𝒍

Theorem (3): First Shifting Rule L{ 𝒆𝒂𝒕 f (t) } = 𝑭 (𝒔−𝒂) Example: Find The Following L { 𝒆𝒂𝒕 cos ⍵t } (d) L { 𝒕 𝒆𝒂𝒕 sin ⍵t }

3 𝑠 2 +9 𝑠 𝑠 2 +25

L{ 𝒕 𝒏 f (t) } = (−𝟏) 𝒏 𝒅 𝒏 𝒅 𝒔 𝒏 𝑳 𝒇 𝒕 Summary L{ 𝒕 𝒏 f (t) } = (−𝟏) 𝒏 𝒅 𝒏 𝒅 𝒔 𝒏 𝑳 𝒇 𝒕 n = ….. L f t Get (−𝟏) 𝒏 𝒅 𝒏 𝒅 𝒔 𝒏 L{ 𝒆𝒂𝒕 f (t) } = 𝑭 (𝒔−𝒂) a = ….. 𝑠 →𝑠−𝑎

SUMMARY From Pages 70 To 77

Thanks