Understanding Laplace Transforms and Their Applications

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Explore Laplace transforms, their linearity, shifting theorems, and application in solving ODEs. Learn about basic transforms, gamma function, and damped vibrations. Dive into examples and gain insights into the world of Laplace transforms.

  • Laplace Transforms
  • ODEs
  • Integration
  • Shifting Theorems
  • Mathematics
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  1. 1 Chapter 6. Laplace Transforms EMLAB

  2. Solving ODE 2 Solution procedure of ODEs ODE Look-up solution Determine integration constants classification Laplace transform Step 1. The given ODE is transformed into an algebraic equation, called the subsidiary equation. Step 2. The subsidiary equation is solved by purely algebraic manipulations. Step 3. The solution in Step 2 is transformed back, resulting in the solution of the given problem. Fig. 113. Solving an IVP by Laplace transforms EMLAB

  3. 6.1 Laplace transform. Linearity. First shifting theorem (s-shifting) 3 Laplace transform definition: If ?(?) is a function defined for all ? 0, ? ? is the Laplace transform of ?(?). ? ? ? ????, ? ? = ? ? = ? = ? + ?? 0 Integral transform: ? ? = ? ? ?(?,?) ?? 0 kernel Inverse Laplace transform : complex integration ?1+? 1 ? ? = 1? ? ? ? ?????, = 2?? ? = ? + ?? ?1 ? EMLAB

  4. 4 EXAMPLE 1 Laplace Transform Let ? ? = 1 when ? 0. Find ?(?) ? ???? = 1 =1 ?? ?? ? = 1 = ? 0 0 ? ? ? 1 Improper integral : ? ???? = lim ? ??? ? ?? = lim ?? ?? ? 0 0 0 EXAMPLE 2 Laplace Transform of the exponential function : (???) Let ? ? = ??? when ? 0, where a is a constant. Find ?(?) 1 1 ???= ???? ???? = ? ?? (? ?)? = when ? ? 0 ? ? 0 0 EMLAB

  5. 5 THEOREM 1 Linearity of the Laplace Transform ?? ? + ??(?) = ? ? ? + ? ? ? EXAMPLE 3 Application of Theorem 1: Hyperbolic Functions Find the transforms of cosh??,sinh?? cosh?? =1 =1 1 1 ? 2 ???+ ? ?? ? ?+ = ?2 ?2 2 ? + ? sinh?? =1 =1 1 1 ? 2 ??? ? ?? ? ? = ?2 ?2 2 ? + ? EXAMPLE 4 Cos and Sine cos?? =1 =1 1 1 ? 2 ????+ ? ??? ? ??+ = ?2+ ?2 2 ? + ?? sin?? =1 =1 1 1 ? 2? ??? ? ?? ? ?? = ?2+ ?2 2? ? + ?? EMLAB

  6. Basic transforms 6 ??+1? ???? = 1 +? + 1 ??+1= ?? ????+1 ??? ???? ? 0 0 0 ??+1=? + 1 ??=? + 1 ? ? + 1 ! ??+2 ? ?? 1= ? ? Table 6.1 Some Functions (t) and Their Laplace Transforms ( (?)). EMLAB

  7. Gamma function 7 ? ? ? ??? 1 ? ????? =? ? + 1 ??= ??? ???? = = ??+1 ??+1 ? ? 0 0 0 ? ????? = ? ! ? + 1 0 s - shifting: Replacing s by (s-a) in the Transform ???? ? = ? ? ? ? ? ? (? ?)??? = [???? ? ]? ???? = ???? ? ? ? ? = 0 0 EXAMPLE 5 s-Shifting: Damped Vibrations. Completing the Square ? ? ???cos?? = (? ?)2+?2 ? (? ?)2+?2 ???sin?? = EMLAB

  8. Existence and Uniqueness of Laplace Transforms 8 A function ? ? has a Laplace transform if it does not grow too fast, say, if for all ? 0 and some constants M and k it satisfies the growth restriction |? ? | ???? If ? ? is defined and piecewise continuous on every finite interval on the semi-axis ? 0 and satisfies growth restriction condition for all ? 0 and some constants M and k, then the Laplace transforms exists for all s>k. ? ? ??? ? ?? ? ? ? ???? ????? ???? = ? = ? ? 0 0 0 Fig. 115. Example of a piecewise continuous function f (t). EMLAB

  9. 9 Uniqueness. If the Laplace transform of a given function exists, it is uniquely determined. Conversely, it can be shown that if two functions (both defined on the positive real axis) have the same transform, these functions cannot differ over an interval of positive length, although they may differ at isolated points EMLAB

  10. 6.2 Transforms of Derivatives and Integrals. ODEs 10 The Laplace transform is a method of solving ODEs and initial value problems. The crucial idea is that operations of calculus on functions are replaced by operations of algebra on transforms. Derivatives 1 ? ? = ? [? ? ] ?(0) ? ? = [? ? ] + ? ? ? ? ???? = ? ??? ? ? ? ? ???? = ?? ? ? 0 ? ? = 0 0 0 ? = ? ??,? = ? ? ? = ? ? ??,? = ?(?) ?? ?? = ?? ? ??? = ?2 ? ?? 0 ? (0) 2 ? ? ? 0 = ? ? ? ? 0 ? 0 ? ? = ? ? ? = ?2 ? ?? 0 ? 0 EMLAB

  11. 11 EXAMPLE 1 Transform of a Resonance Term Let ? ? = ?sin??. Find ?(?) ? 0 = 0, ? ? = sin?? + ??cos??,? 0 = 0 ? ? = 2?cos?? ?2?sin?? ? ? = 2? ?2+ ?2 ?2 ? = ?2 (?) ? ? = ?sin?? = 2? ?2+ ?2 2 EMLAB

  12. THEOREM 3 : Laplace Transform of the Integral of a Function 12 ? ? ? ? ?? =1 ? ? ?? = 11 ?? ? , thus ?? ? 0 0 Proof : ?? ?? = ?? ? ??? ? ? ? ? ?? ? ???? = 1 +1 ?(?)? ???? =1 ?? ?? ? ? ?? ? ??(?) 0 0 0 0 0 ? ? ? ??,? = ? ?? ? = ? ? ,? = 1 ?? ?? ? = 0 EMLAB

  13. 13 EXAMPLE 3 Application of Theorem 3: 1 1 Find the inverse of ?(?2+?2), ?2(?2+?2) ?2+ ?2=sin?? 1 1 , ? ?sin?? ? 1 1 1 = ?? = ?2(1 cos??) ?(?2+ ?2) 0 ? 1 1 ?2? sin?? 1 1 = ?2 (1 cos??) ?? = ?2(?2+ ?2) ? 0 EMLAB

  14. Differential Equations, Initial Value Problems 14 ? + ?? + ?? = ? ? , ? 0 = ?1 ? 0 = ?0, Step 1. Setting up the subsidiary equation. Applying Laplace transforms on both sides, ?2? ? ?? 0 ? 0 + ? ?? ? ? 0 + ?? ? = ?(?) Step 2. Solution of the subsidiary equation by algebra. ?2+ ?? + ? ? ? = ? + ? ? 0 + ? 0 + ?(?) ? + ? ? 0 + ? 0 + ?(?) ?2+ ?? + ? ? ? = Step 3. Inversion of Y to obtain ? ? = 1[? ? ] EMLAB

  15. 15 EXAMPLE 4 Initial Value Problem: The Basic Laplace Steps Solve ? ? ? ? = ?, ? 0 = 1 ? 0 = 1, Solution: 1 ?2 (?2 1)? ? = ? + 1 +1 ?2? ?? 0 ? 0 ? = ?2 ? + 1 +1 ?2 1 ?2 1 1 1 ? ? = = ? 1+ ?2 1 ?2 1 1 1 ?2 ? ? = 1? ? = 1 + 1 1 ?2 1 ? 1 = ??+ sinh? ? EMLAB

  16. 16 Fig. 116. Steps of the Laplace transform method EMLAB

  17. 17 EXAMPLE 5 Comparison with the Usual Method Solve the initial value problem ? + ? + 9? = 0 , ? 0 = 0 ? 0 = 0.16, Solution: ?2? ?? 0 ? 0 + ?? ? 0 + 9? = 0 (?2+ ? + 9)? ? = 0.16(? + 1) 0.16 ? +1 2+ 0.08 2 +35 ? ? =0.16 ? + 1 ?2+ ? + 9= ? +1 2 4 35 4? +0.08 35 4? ? ? = 1? = ? ? 0.16cos sin 2 35 4 = ? 0.5?(0.16cos2.96? + 0.027sin2.96?) EMLAB

  18. 6.3 Unit step function, second shifting theorem (t-shifting) 18 ? ? = 1 (? > 0) (? < 0) ? ? ? = 1 (? > ?) (? < ?) Unit step function : 0 0 ? ? ? ???? = 1 =1 ?? ?? ? ? = ? 0 0 =? ?? ? ???? = 1 ? ? ? ? ???? = ?? ?? ? ? = ? ? 0 ? EMLAB

  19. 19 Fig. 120. Effects of the unit step function: (A) Given function. (B) Switching off and on. (C) Shift. Pulse function EMLAB

  20. 20 THEOREM 1 Second Shifting Theorem; Time Shifting If ?(?) has the transform ?(?), then the shifted function 0 (? < ?) (? > ?) ? ? = ?(? ?)? ? ? = ?(? ?) has the transform ? ???(?) {? ? ? ? ? ? } = ? ???(?) ? ? ? ? ? ? = 1{? ???(?)} ? ? ? ?(? ?)? ???? = ? ? ? ?(?+?)?? = ? ???(?) 0 0 ? = ? ? EMLAB

  21. 21 EXAMPLE 1 Application of Theorem 1. Use of Unit Step Functions Write the following function using unit step functions and find its transform. 2 if 0 < ? < 1 if 1 < ? <? if ? >? 1 2?2 ? ? = 2 cos? 2 Solution. +1 2?2? ? 1 ? ? ? + (cos?)? ? ? ? ? = 2 1 ? ? 1 2 2 1 ? ? ? 2 1 ? ? 1 = 2 1 2?2? ? 1 1 2(? 1)2? ? 1 1 ?3+1 ?2+1 ? ? = = 2? EMLAB

  22. 22 2?2+?2 2 1 2?2? ? ? 1 2 ? ? ? ? ? 1 ?3+ ? ? ??/2 = = 2 2 2 8? (cos?)? ? ? = cos ? ? ? ? ? 1 ?2+ 1? ??/2 = 2 2 2 1 ? ? ? 2?2+?2 1 ?3+1 ?2+1 1 ?3+ ? 1 ? ? ? ??/2 ?2+ 1? ??/2 ? = 2 + 2? 8? Fig. 122. (t) in Example 1 EMLAB

  23. 23 EXAMPLE 3 Response of an RC-Circuit to a Single Rectangular Wave Find the current ?(?) in the RC-circuit in Fig. 124 if a single rectangular wave with voltage ?0is applied. The circuit is assumed to be quiescent before the wave is applied. Fig. 124. RC-circuit, electromotive force ?(?) ? ?? ? +? ? = ?? ? +1 ? ? ? ?? = ? ? = ?0[? ? ? ? ? ? ] ? 0 EMLAB

  24. 24 ?? ? +? ? =?0 ?[? ?? ? ??] ?? 1 ?? ? ? =?0 ?[? ?? ? ??] ? + ?0 ? ?0/? ? +1 ? ?? ? ??= [? ?? ? ??] ? ? = ? +1 ?? ?? ? ? = 1[? ? ] =?0 exp ? ? ? ? ? exp ? ? ? ? ? ? ?? ?? EMLAB

  25. 6.4 Short Impulses, Diracs Delta Function, Partial Fractions 25 ??? ? = 1/? if ? ? ? + ? otherwise 0 ?+?1 ??= ??? ? ?? = ??? = 1 0 ? Fig. 132. The function ??? ? Dirac delta function : ? ? ? = lim ? 0??(? ?) ? ? ? = if ? = ? otherwise 0 ? ? ? ?? = 1 0 EMLAB

  26. 26 Sifting property of delta function ?2 ?(?)? ? ?0?? = ?(?0) ?0 [?1,?2] ?0 [?1,?2] 0 ?1 Laplace transform of delta function ??? ? =1 ?[? ? ? ? ? ? ? ] 1 ??? ?? ? ?+? ?= ? ??1 ? ?? ??? ? = ?? We now take the limit as ? 0, = ? ?? ? ? ? Laplace transform of delta function ?(?)? ???? = 1 ? ? = 0 ?(? ?)? ???? = ? ?? ? ? = 0 EMLAB

  27. 27 EXAMPLE 2 Hammer-blow Response of a Mass Spring System Determine the response of the damped mass spring system under a unit impulse at time ? = 1, ? + 3? + 2? = ?(? 1) Solution. ?2+ 3? + 2 ? ? = ? ? ? ? 1 1 ? ? ? ? = = ? + 1 ?2+ 3? + 2 ? + 2 0 if 0 ? 1 if ? > 1 ? ? = 1? ? = ? ? 1 ? 2 ? 1 Fig. 134. Response to a hammer-blow in Example 2 EMLAB

  28. Inverse transform 28 =????+ ?? 1?? 1+ + ?1? + ?0 ????+ ?? 1?? 1+ + ?1? + ?0 ? ? =? s (??,?? real number) ? s ? s ? s = ?? ??? ?+ + ?2?2+ ?1? + ?0+?1(?) for ? > ? ?(?) (1) If ? ? has simple roots, ?1(?) ?(?)= ?1 ?2 ?? + + + ? + ?1 ? + ?2 ? + ?? (2) If ? ? has simple complex poles, ?1(?) ?1 ?1 ?1(?)(? + ? ??)(? + ? + ??)= ? + ? ??+ ? + ? + ??+ (3) If ? ? has a root of multiplicity r, ?1(?) ?11 ? + ?1 ?12 ?1? 2= + 2+ + ?+ ?1? ? + ?1 ? + ?1 ? + ?1 EMLAB

  29. Partial fraction expansion 29 (1) Simple poles ? ? =?(?) ?1 ?2 ?? ?(?)= + + + ? + ?1 ? + ?2 ? + ?? ? + ??? ? ? ? ? + ??? ? = = 0 + + 0 + ??+ 0 + + 0 ?= ?? ?= ?? ?? 1 = ?1? ???(?) ? + ? (2) Complex conjugate poles ?1(?) ?1 ?1 ? ? = ?1(?)(? + ? ??)(? + ? + ??)= ? + ? ??+ ? + ? + ??+ ? + ? ?? ? ? ?= ?+??= ?1 ? (?+??)?+ ? ???+ 1?(?) = ?1? (? ??)?+ ?1 = ? ???1????+ ?1 = 2 ?1? ??cos(?? + ?1) + EMLAB

  30. 30 (3) Multiple poles ?1(?) ?1,1 ? + ?1 ?1,2 ? + ?1 ?1,? ? + ?1 ? ? = 2= + 2+ + ?+ ?1? ? + ?1 ?? ? ? + ?1 ?= ?1= ?1,? ? ??[ ? + ?1 ?? ? ] = ?1,? 1 ?= ?1 ?2 ??2[ ? + ?1 ?? ? ] = (2!)?1,? 2 ?= ?1 ?? ? ??? ?[ ? + ?1 1 ?? ? ] ?1,?= ? ? ! ?= ?1 ? + ??+1=?? 1 1 ?!? ?? EMLAB

  31. Example 31 Let s determine the inverse Laplace transform of the following function. 10(? + 3) ? + 13(? + 2)= ?11 ? + 1+ ?12 ? + 12+ ?13 ? + 13+ ?2 ? ? = ? + 2 ? + 13 ? + 2 ? + 13? ? = ?11? + 12+ ?12? + 1 + ?13+ ?2 ? + 13? ? ?13= ?= 1= 20 ? ?? ? ?? 10 ? + 3 ? + 2 10 ? + 13? ? ?12= = = ? + 22= 10 ?= 1 ?2 ??2 ?2 ??2 ?11=1 =1 10 ? + 3 ? + 2 10 ? + 13? ? = ? + 23= 10 2! 2 ?= 1 ?2= ? + 2 ? ? ?= 2= 10 10 10 20 10 ? + ??+1=?? 1 ? ? = ? + 1 ? + 12+ ? + 13 1 ?!? ?? ? + 2 ? ? = [(10 10? + 10?2)? ? 10? 2?]?(?) EMLAB

  32. 6.5 Convolution. Integral Equations 32 Definition: ? ? ? ? ? ? ? ? ? ? ? ?? = ? ? ? ?(?)?? 0 0 ? ? ? ?(? ?)?? = ? ? ? ? = ? ? ?(?) 0 ? ? ? ?(? ?)?? ? ???? 0 0 ? ? ? ? ? ?(? ?)?? ? ???? = 0 0 ? ? ? ?(? ?)? ???? ?? = ? ? 0 0 ? ? ? ???? ?(?) = 0 = ? ? ?(?) EMLAB

  33. 33 Properties of convolution (commutative law) ? ? = ? ? (distributive law) ? ?1+ ?2 = ? ?1+ ? ?2 (associative law) ? ? ? = ? ? ? ? 0 = 0 ? = 0 EXAMPLE 2 Convolution 1 Find (?). ? ? = ?2+ ?2 2 Solution. ? sin?? = ?2+ ?2 ? ? =sin?? sin?? ? 1 = ?2 sin??sin?(? ?)?? ? 0 EMLAB

  34. 34 ? 1 = 2?2 cos?? + cos 2?? ?? ?? 0 2?2 ?cos?? +sin?? 1 = ? EXAMPLE 3 Unusual Properties of Convolution ? 1 ? in general ? ? 1 ?? =1 2?2 ? ? 1 = 0 ? ? 0 may not hold. sin? sin? = 1 2?cos? +1 2sin? Fig. 142. Example 3 EMLAB

  35. 35 EXAMPLE 4 Repeated Complex Factors. Resonance 2? = ? sin?0?, ? + ?0 ? 0 = 0 ? 0 = 0, ??0 ?2+ ?0 ??0 ? ?0 ?0 ?0 2? = ?2+ ?0 2 ? ? = 2 2= 2 2 ?2+ ?0 ?2+ ?0 ?2+ ?0 ? ?0 ? ? ? = sin?0? sin?0? = 2 ?0?cos?0? + sin?0? 2?0 EMLAB

  36. Application to Nonhomogeneous Linear ODEs 36 ? + ?? + ?? = ? ? , ?,? constant ?2? ? ?? 0 ? 0 + ? ?? ? ? 0 + ?? ? = ?(?) ?2+ ?? + ? ? ? = ? + ? ? 0 + ? 0 + ?(?) 1 ? + ? ? 0 + ? 0 ? ? = ? ? + ? ? ? ? , ? ? = ?2+ ?? + ? ? ?? ? ??? = ? ? ? ?(?)?? 0 ? ? = ?1? ?1?+ ?2? ?2? EMLAB

  37. 37 EXAMPLE 5 Response of a Damped Vibrating System to a Single Square Wave Using convolution, determine the response of the damped mass spring system modeled by ? ? = 1 1 < ? < 2 otherwise ? + 3? + 2? = ? ? , 0 ? 0 = 0,? 0 = 0 Solution by Convolution : 1 1 1 1 ? ? = ?2+ 3? + 2= (? + 1)(? + 2)= ? + 1 ? + 2 ? ? = ? ? ? 2??(?) ? ? = ? ? ? 1 ?? = [? (? ?) ? 2(? ?)] ?? = ? (? ?) 1 2? 2(? ?) EMLAB

  38. 38 ? ? ? = ? ? ? ? ? 1 ? ? 2 ?? 0 ? 1 2 0 ? < 1 ? ? = 0 1 2 0 ? 1 < ? < 2 1 2 0 ? ? ? ? = ? ? ? 1 =1 2 ? ? 1+1 2? 2 ? ? 2? 2(? 1) 1 ? > 2 ? 1 2 0 2 ? ? = ? ? ? 1 = ? ? 2 1 2? 2(? 2) ? ? 1+1 2? 2 ? ? 2? 2(? 1) 1 EMLAB

  39. 39 Fig. 143. Square wave and response in Example 5 EXAMPLE 6 A Volterra Integral Equation of the Second Kind Solve the Volterra integral equation of the second kind ? ? ? ? ? sin ? ? ?? = ? 0 Solution. ? ? sin? = ? ?2 1 1 ?2 ? ? = 1 ?2+1 ? ? ? ? ?2+ 1= ? ? ?2+ 1= ?4 ? ? = ? +?3 6 EMLAB

  40. 6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients 40 Differentiation of Transforms ? ? ? ???? ?? ??= ? ? = [?? ? ]? ???? ? ? = 0 0 = ? ? , 1? ? ?? ? = ??(?) EXAMPLE 1 Differentiation of Transforms ?sin?? = ? ? 2?? ?2+ ?2= ?2+ ?2 2 ?? ?2+ ?2 2?2 ?2+ ?2 2 ?2 ?2 ?2+ ?2 2 ?cos?? = ? ? ?2+ ?2= = ?? EMLAB

  41. 41 Integration of Transforms ? ? ? ???? ? ? = ? ??? ? ?? ? ? ? ? = ? ? ? ? 0 0 ? ? ? 1 ? ? ?? ???? = = ? 0 ? ? ? ? ? ? =? ? 1 = ? ? ? ?, ? ? ? ? EXAMPLE 2 Differentiation and Integration of Transforms ln 1 +?2 = ln?2+ ?2 Find the inverse transform of ?2 ?2 ?2+ ?2 2? 2? ? ? = ?2 ?2+ ?2 2? ? ? =2 2? 1? ? = 1 ?2= 2cos?? 2 = ?? ? ?(1 1cos??) EMLAB

  42. Special Linear ODEs with Variable Coefficients 42 ?? = ? = ? ??? ???? ? 0 ?? ?? = ? = 2?? ?2?? ???2? ?? 0 ? 0 ??+ ?(0) EXAMPLE 3 Laguerre s Equation. Laguerre Polynomials ?? + 1 ? ? + ?? = 0 2?? ?2?? + ?? ? 0 ? ??? ??+ ? 0 + ?? = 0 ?? ? ?2?? ??+ ? + 1 ? ? = 0 ?? ?= ? + 1 ? ? 1 ? + 1 ? ?? = ?? ? ?2 ? ? 1? ??+1 ? 1? ??+1 ln? = ln ? = EMLAB

  43. 43 =?? ?? ?????? ?, ??? = 1? ? ? = 0,1,2,3, ?! ?? ?????? ? ?!?? ? + 1?+1 ?! ??? ?= ? + 1?+1, = EMLAB

  44. 6.7 Systems of ODEs 44 The Laplace transform method may also be used for solving systems of ODEs. = ?11?1+ ?12?2+ ?1(?) = ?21?1+ ?22?2+ ?2(?) ?1 ?2 ??1 ?1(0) = ?11?1+ ?12?2+ ?1(?) ??2 ?2(0) = ?21?1+ ?22?2+ ?2(?) (?11 ?)?1+ ?12?2= ?10 ?1(?) ?21?1+ ?22 ? ?2= ?20 ?2(?) ? ?? ?1 ?2 ?1 ?2 ?1 ?2 ?11 ?21 ?12 ?22 = + ?11 ? ?21 ?12 = ?10 ?1 ?2 ?1 ?22 ? ?2 ?20 EMLAB

  45. EXAMPLE 2 Electrical Network 45 Find the currents ?1? ,?2(?) when ? 0 = 0,? 0 = 0 ? ? = 100 0 ? 0.5 otherwise 0 Solution : ??1 ??+ ?1?1 ?2 + ?2?1= 100[1 ? ? 0.5 ] ?1 ??2 ??+ ?1?2 ?1 = 0 ?2 ? ? ? 1 2 0.8??1+ 1 ?1 ?2 + 1.4?1= 100 ??2+ 1 ?2 ?1 = 0 ? 125 ? + 1 ? ? +1 125 1 ? ? 1 ? ? ?1= 2 , ?1= 2 ? +7 ? ? +1 ? +7 2 2 2 2 ?1? = 125 ?2? = 250 2 625 2 +500 ? ? 21? 7? 21? 7? 3 7 2+250 2 +500 ? ? EMLAB 3 7

  46. Properties of the Laplace transform 46 EMLAB

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