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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)