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CHAPTER 4. The Laplace Transform. Contents. 4.1 Definition of the Laplace Transform 4.2 The Inverse Transform and Transforms of Derivatives 4.3 Translation Theorems 4.4 Additional Operational Properties 4.5 The Dirac Delta Function. 4.1 Definition of Laplace Transform. DEFINITION 4.1.

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## CHAPTER 4

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**CHAPTER 4**The Laplace Transform**Contents**• 4.1 Definition of the Laplace Transform • 4.2 The Inverse Transform and Transforms of Derivatives • 4.3 Translation Theorems • 4.4 Additional Operational Properties • 4.5 The Dirac Delta Function**4.1 Definition of Laplace Transform**DEFINITION 4.1 If f(t)is defined for t 0,then (2)is said to be the Laplace Transform of f. Laplace Transform • Basic DefinitionIf f(t)is defined for t 0,then the improper integral(1)**Example 1**Evaluate L{1} Solution: Here we keep that the bounds of integral are 0 and in mind.From the definition , s>0Since e-st 0as t ,for s > 0.**Example 2**Evaluate L{t} Solution , s>0**Example 3**Evaluate L{e-3t} Solution**Example 4**Evaluate L{sin2t} Solution**Example 4 (2)**Laplace transform of sin 2t ↓**L.T. is Linear**• We can easily verify that(3)**THEOREM 4.1**(a) (b) (c) (d) (e) (f) (g) Transform of Some Basic Functions**DEFINITION 4.2**A function f(t)is said to be of exponential order, if there exists constants c>0, M > 0, and T > 0,such That |f(t)| Mect for all t > T. See Fig 4.2, 4.3. Exponential Order**Fig 4.4**• A function such as is not of exponential order, see Fig 4.4**THEOREM 4.2**If f(t) is piecewise continuous on [0, ) and of exponential order, then L{f(t)}exists for s > c. Sufficient Conditions for Existence**Example 5**Find L{f(t)}for Solution**4.2 If F(s)=L(f(t)), then f(t) is the inverse Laplace**transform of F(s) and f(t)=L(F(s)) THEOREM 4.3 (a) (b) (c) (d) (e) (f) (g) Some Inverse Transform**Example 1**Find the inverse transform of (a) (b) Solution(a)(b)**L -1 is also linear**• We can easily verify that(1)**Example 2**Find Solution(2)**Example 3: Partial Fraction**Find SolutionUsing partial fractionsThen (3)If we set s = 1, 2, −4,then**Example 3 (2)**(4)Thus (5)**Uniqueness of L -1**• Suppose that the functions f(t) and g(t) satisfy the hypotheses of Theorem 4.2, so that their Laplace transform F(s) and G(s) both exist. If F(s)=G(s) for all s>c (for some c), then f(t)=g(t) whenever on [0, + ) both f and g are continuous.**Transform of Derivatives**, s>0 (6) , s>0 (7) (8)**THEOREM 4.4**If are continuous on [0, ) and are of Exponential order and if f(n)(t) is piecewise-continuous On [0, ), thenwhere Transform of a Derivative**Solving Linear ODEs**• Then(9)(10)**Example 4: Solving IVP**Solve Solution(12)(13)**Example 4 (2)**We can find A = 8, B = −2, C = 6Thus**Example 5**Solve Solution(14)Thus**4.3 Translation Theorems**THEOREM 4.5 Proof If f is piecewise continuous on [0, ) and of exponential order, then limsL{f} = 0. Behavior of F(s) as s → **THEOREM 4.6**ProofL{eatf(t)} = e-steatf(t)dt = e-(s-a)tf(t)dt = F(s – a): replacing all s in F(s) by s-a If L{f} = F(s) and a is any real number, then L{eatf(t)} = F(s – a), See Fig 4.10. Translation on the s-axis**Example 1**Find the L.T. of(a) (b) Solution(a)(b)**Inverse Form of Theorem 4.6**• (1)where**Parttial Fraction**• To perform the inverse transform of R(s)=P(s)/Q(s): • Rule 1: Linear Factor Partial Fractions • Rule 2: Quadratic Factor Partial Fractions**Example 2**Find the inverse L.T. of(a) (b) Solution(a) we have A = 2, B = 11 (2)**Example 2 (2)**And(3)From (3), we have(4)**Example 2 (3)**(b) (5) (6) (7)**Example 3**Solve Solution**Example 3 (2)**• (8)**Example 4**Solve Solution**DEFINITION 4.3**The Unit Step Function U(t – a) defined for is Unit Step Function See Fig 4.11.**Fig 4.12 Fig 4.13**• Fig 4.12 shows the graph of (2t – 3)U(t – 1).Considering Fig 4.13, it is the same as f(t) = 2 –3U(t – 2) +U(t – 3),**Also a function of the type(9)is the same as(10)Similarly, a**function of the type(11)can be written as (12)**Example 5**Express in terms of U(t). See Fig 4.14. SolutionFrom (9) and (10), with a = 5, g(t) =20t, h(t) = 0 f(t) =20t – 20tU(t – 5)

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