Transforms and its Applications: UNIT 1: Laplace Transforms

Initial and Final Value Theorems

Laplace Transforms

Laplace Transforms: Example Important Solved Problems with formula, steps, derivation and answer based on Initial and Final Value Theorems

INITIAL AND FINAL VALUE THEOREMS

 

INITIAL VALUE THEOREM


Proof: We know that,

 L[f '(t)] = sL[f(t)] ‒ f(0) = s F(s) ‒ f(0)


 

FINAL VALUE THEOREM


*Note: The author wishes to thank Dr. B. Jothiram, formerly Asst. Prof. and H.O.D. Maths (Retd.) Govt. College of Engineering, Salem who has suggested to him the block diagram.

 

PROBLEMS BASED ON INITIAL VALUE AND FINAL VALUE THEOREMS

 



Example 2: Verify the initial and final value theorem for the function f(t) = 1 + e‒1(sin t + cos t)

Solution:


Hence, Initial value theorem verified.

Final value theorem states that


L.H.S = R.H.S

Hence, Final value theorem verified.

 

Example 3: Verify the initial and final value theorems for f(t) = 3 e‒2t

Solution:

 f(t) = 3 e‒2t


 L.H.S = R.H.S

Hence, Initial value theorem verified.

Final value theorem


 L.H.S. = R.H.S.

Hence, Final value theorem verified.

 

Example 4 Using the initial value theorem, find


Solution:

Initial value theorem states that


 

Transforms and its Applications: UNIT 1: Laplace Transforms : Tag: Engineering mathematics, Maths : Laplace Transforms - Initial and Final Value Theorems


Transforms and its Applications: UNIT 1: Laplace Transforms



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