Chapter 3.pdf - ENGI 3424 3 3 \u2013 Laplace Transforms Page 3.01 Laplace Transforms In some situations a difficult problem can be transformed into an

# Chapter 3.pdf - ENGI 3424 3 3 – Laplace Transforms Page...

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ENGI 3424 3 – Laplace Transforms Page 3.01 3. Laplace Transforms In some situations, a difficult problem can be transformed into an easier problem, whose solution can be transformed back into the solution of the original problem. For example, an integrating factor can sometimes be found to transform a non-exact first order first degree ordinary differential equation into an exact ODE [Section 1.7]: One type of first order ODE [not considered in this course] is the Bernoulli ODE, y' + P y = R y u , which, upon a transformation of the dependent variable, becomes a linear ODE: An initial value problem is an ordinary differential equation together with sufficient initial conditions to determine all of the arbitrary constants of integration. A Laplace transform will convert some initial value problems into much easier algebra problems. The solution of the original problem is then the inverse Laplace transform of the solution to the algebra problem. Uses of Laplace transforms include: 1) the solution of some ordinary differential equations 2) the solution of some integro-differential equations, such as ( ) ( ) ( ) ( ) 0 t y t g t h t x y x d = + x .
ENGI 3424 3 – Laplace Transforms Page 3.02 Contents : 3.01 Definition, Linearity, Laplace Transforms of Polynomial Functions 3.02 Laplace Transforms of Derivatives 3.03 Review of Complex Numbers 3.04 First Shift Theorem, Transform of Exponential, Cosine and Sine Functions 3.05 Applications to Initial Value Problems 3.06 Laplace Transform of an Integral 3.07 Heaviside Function, Second Shift Theorem; Example for RC Circuit 3.08 Dirac Delta Function, Example for Mass-Spring System 3.09 Laplace Transform of Periodic Functions; Square and Sawtooth Waves 3.10 Derivative of a Laplace Transform 3.11 Convolution; Integro-Differential Equations; Circuit Example Note : Sections 3.09, 3.10 and the latter part of 3.11 will be parts of this course only if time permits.
ENGI 3424 3.01 – Definition; Polynomial Functions Page 3.03 3.01 Definition, Linearity, Laplace Transforms of Polynomial Functions If f ( t ) is defined on t > 0, piece-wise continuous on t > 0 (that is, only a finite number of finite discontinuities) and of exponential order [ | f ( t ) | < k e α t for all t > 0 and for some positive constants k and α ], then exists and ( ) ( ) 0 lim m m st F s e f t dt → ∞ = the Laplace transform of f ( t ) is ( ) ( ) ( ) { } 0 st F s e f t dt f t = = L Example 3.01.1 Find L { 1 } .
ENGI 3424 3.01 – Definition; Polynomial Functions Page 3.04 Example 3.01.2 Find L { t } . Linearity Property of Laplace Transforms L { a f ( t ) + b g ( t ) } (where a , b are constants):
ENGI 3424 3.01 – Definition; Polynomial Functions Page 3.05 Example 3.01.3 Find L { 2 t 3 } . Example 3.01.4 Find L { t n } .