Laplace Transforms PDF

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Chapter(7(Laplace(Transforms( ( 7.1(Introduction:(A(Mixing(Problem( !

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! Notice!that!the!input!rate!is!a!piecewise!defined!function:! ! ! ! g(t)!=! ! ! Write!the!IVP!in!terms!of!g(t):! ! ! ! ! !This!would!be!time-consuming!to!solve!using!the!methods!of!Chapter!4:! ! ! ! ! ! ! !

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In!Chapter!7!we!will!use!the!Laplace(transform!(with!the!help!of!transform!tables)! to!make!a!change!in!variable!that!will!transform!the!differential!equation!into!an! algebraic!equation,!solve,!and!then!apply!an!inverse!transformation!to!find!the! solution!to!the!IVP.!It!should!not!be!surprising!that!a!Laplace!transform!will!be! defined!in!terms!of!an!integral,!which!will!include!a!change!in!variable!from!t!to!s.! ! Laplace!transform!of!!x(t)!!(t-domain)!is!!X(s)!(s-domain).!Transformed!algebraic! equation!(s-domain):! ! ! ! ! ! Solve!for!X(s),!apply!inverse!Laplace!Transform!(from!table)!to!obtain!solution,!x(t).! !

! ( 7.2(Definition(of(Laplace(Transform( ( For!a!function! f (t ) !!on![0, ! ∞) !,!the!Laplace(transform!of!! f (t ) !is!the!function!!!F(s) ! ! ∞ given!by! F(s) = L { f (t )}(s) = ∫ e − st f (t )dt .!!! 0 ! ! The!domain!is!the!set!of!all!s!for!which!the!improper!integral!exists.! ! ∞

N

More!precisely:! ∫ e − st f (t )dt = lim ∫ e − st f (t )dt ! 0

N→∞ 0

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Example!1:!Find!the!Laplace!transform!of! f (t ) = 1 !and!give!the!domain.! ! ! ! ! ! ! ! ! Example!2:!Find!the!Laplace!transform!of!! f (t ) = e3t .!!Generalize!and!give!the! domain.! ! ! ! ! ! ! ! ! ! ! ! ! ! Example!3:!!Find! L {sin4t } .!!Generalize!and!give!the!domain.! ! ! ! ! ! ! ! ! ! Linearity( ! Theorem(1.!!Let!f,!!f1!and!f2!be!functions!that!have!Laplace!transforms!for! s > α !!and! let!c!!be!a!constant.!!Then! ! ! (a)! ! ! ! (b)! ! ! !

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Example!4:!Find!the!Laplace!transform!of!! ⎧ 3, 0 < t < 4 ⎪ f (t ) = ⎨ 0, 4 < t < 6 !!! ⎪ e2t , t >6 ⎪⎩ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Example!5:!Find! L {3 − 2sin3t + 6e2t } ! ! ! ! ! ! ! ! ! Existence(of(the(Transform( ! (1)!!A!function!is!piecewise(continuous!on![a, b] !!if!it!continuous!at!every!point!in! the!interval!except!for!a!finite!number!of!jump!discontinuities.! ! (2)!!A!function!is!of!exponential(order!if!it!doesn’t!grow!faster!than!an!exponential! function.! ! Theorem(2.!!If! f (t ) !!is!piecewise!continuous!on![0, ∞) !!and!of!exponential!order,! then!the!Laplace!transform!exists!for! !s > α !.! !

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! ! Example!6:!Find!!!L {3t 3 − 2e 4t cos3t + 6te2t }! ! ! ! ! Use!the!table!of!Laplace!transforms!to!make!a!conjecture!about! lim F(s) .!!! s→∞

! 7.2!Group!Exercises! ! ∞

1.!!Use!the!definition! F(s) = L { f (t )}(s) = ∫ e − st f (t )dt !to!find!!!L {t } .! 0 ! ! ! ! ! ! ! ! ! ! !

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2.!Use!the!definition! F(s) = L { f (t )}(s) = ∫ e − st f (t )dt !to!find!!L { f (t )} !where! 0 ! ⎧⎪ 0, 0 ≤ t < 3 f (t ) = ⎨ !!! 2, t ≥3 ⎪ ⎩ ! ! ! ! ! ! ! ! ! ! ! ! ! 7.3(Properties(of(the(Laplace(Transform( ( Translation(in(s( ( Theorem(3.((If!the!Laplace!transform!of!f(t)!=!F(s)!exists!for!some! s > α ,!then!! ! L e at f (t ) (s) = !!!!!! ! ! ! for!s!>! ! ! ! ! ! ! ! ! ! Example!1:!!Find!the!Laplace!transform!of! e3t cos4t !using!the!translation!property.! ! ! ! ! ! ! ! ! ! ! ! !

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Laplace(Transform(of(the(Derivative( Theorem(4.((!Let!f(t)!be!continuous!on![0, ∞) !!and!f'‘(t)!be!piecewise!continuous!on! [0, ∞) ,!with!both!of!exponential!order! α !.!!Then!for! s > α ,! ! L f '(t ) (s) = !! ! ! ! ! proof:! ! ! ! ! ! ! ! Laplace(Transform(of(Higher-Order(Derivatives( Theorem(5.((!Let!! f (t ), f '(t ) !be!continuous!on![0, ! ∞) !!and!f'”(t)!be!piecewise! continuous!on! [0, ∞) ,!with!both!of!exponential!order! α !.!!Then!for! s > α ,! ! L f "(t ) (s) = !!

{

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{

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! ! Derivatives(of(the(Laplace(Transform(( Theorem(6.((!Let!f(t)!be!piecewise!continuous!and!exponential!order! α !on! [0, ! ∞) ,!

{

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with! F(s) = L f (t ) .!!Then!for! s > α ,! ! L t n f (t ) (s) = !!

{

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! proof:! ! ! ! ! ! ! ! ! ! ! ! Example!2:!!Find! L t sinbt .!

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! ! ! ! ! ! ! ! ! Example!3:!!Find!the!Laplace!transform!of!each.! ! (a)!(1 + e −t )3 !! ! ! ! ! (b)!! t cos2 t !! ! ! ! ! ! ! ! 7.4(Inverse(Laplace(Transforms( ( After!applying!Laplace!transforms!to!a!differential!equation,!we!will!need!to!perform! an!inverse!Laplace!transform!to!find!the!solution!to!the!differential!equation.!!! ! Apply!Laplace!transforms!to!the!differential!equation!IVP:!! y"− y = −t;

y(0) = 0,

y '(0) = 1 !!!

! ! ! ! Now,!solve!for!Y(s)!and!simplify.! ! ! ! ! ! We!need!to!find!the!function!y(t)!that!corresponds!to!this!Y(s).!!What!is!it?! ! Given!a!function!F(s),!if!there!is!a!function!f(t)!that!is!continuous!on! [0, ∞) !where!

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L f (t ) = F(s) !!then!f(t)!is!the!inverse(Laplace(transform!of!F(s)!and!we!write! ! ! f(t)!=!! ! Example!1:!!Find!the!inverse!Laplace!transform!of!each.! ! 6 (a)!! F(s) = 4 !!! s ! 4 (b)!! F(s) = 2 !! s +16 !! ! s +1 (c)!! F(s) = 2 !!! s + 2s +10 !! ! ! ( ( Linearity(of(the(Inverse(Transform( ( L −1 F1 + F2 = (((

{

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( L −1 cF = (( ! !

{ }

3 ⎫ −1 ⎧ Example!2:!!Find! L ⎨ !!! 3 ⎬ ⎩ (s − 4) ⎭ !! ! ! ! ⎧ 3s + 2 ⎫ Example!3:!Find!L −1 ⎨ 2 ⎬ .!!Hint:!complete!the!square.! + 2s +10 s ⎭ ⎩ ! ! ! ! ! ! ! !

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⎫ ⎧ 7s −1 Example!4:!!Find! L −1 ⎨ ⎬ .!!Use!partial!fraction!decomposition.!! ⎩ (s +1)( s + 2)( s − 3) ⎭ !! ! ! ! ! ! ! ! ! ! ! ! ! ! ⎧ s 2 + 9s + 2 ⎫ Example!5:!Find!L −1 ⎨ ⎬! (s −1)2(s + 3) ⎪⎭ ⎪ ⎩ !! ! ! ! ! ! ! ! ! ! ! ! ! ⎧ 6s 2 +50 ⎫ Example!6:!!Find! L −1 ⎨ ⎬! (s + 3)( s 2 + 4) ⎪⎭ ⎪ ⎩ !! ! ! ! ! ! ! ! ! ! ! ! !

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⎫ 2s 2 +10s −1 ⎧ Example!7:!!Find! L ⎨ 2 ⎬! (s − 2s + 5)( s + 1) ⎭⎪ ⎪ ⎩ !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 7.4!Group!Exercises! ! s2 + 4 1.!!Solve!for!F(s).!! s 2F (s)+ sF(s)− 6F (s) = 2 !! s +s !! ! ! ! 2.!!Determine!the!form!of!the!partial!fraction!decomposition!for!F(s)!above.! ! ! ! ! 3.!!Find!the!coefficients!and!find!the!inverse!transform!of!F(s).! ! ! ! ! !

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7.5(Solving(Initial(Value(Problems( ! Laplace!transforms!provide!a!third!method!(with!undetermined!coefficients!and! variation!of!parameters)!for!solving!second-order!differential!equation!IVPs.!!Some! benefits!of!this!new!method!include:!you!don’t!have!to!find!the!homogeneous! solution!first!to!solve!nonhomogeneous!equations,!and!it!can!handle!discontinuous! forcing!functions,!variable!coefficients,!systems!of!differential!equations,!and!partial! differential!equations.!!This!method!is!the!preferred!method!for!many!applications.! ! Method!of!Laplace!transforms!to!solve!a!differential!equation!IVP:! ! 1.! ! 2.! ! ! ! 3.! ! ! ! Example!1:!Use!Laplace!transforms!to!solve!the!IVP:! y"− 2 y'+5 y = −8e −t ; y(0) = 2, y '(0) = 12 !!! ! As!an!intermediate!step,!solve!for!Y(s),!the!Laplace!transform!of!the!solution!y(t).!! Write!your!answer!as!one!simplified!fraction.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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A(Brief(Table(of(Laplace(Transforms( ( F(s) = L f (t ) (s) !!! ! f (t ) !!

{

at !e f (t ) ! f '(t )! f "(t ) ! n !t f (t )! at !e !!

n !t !!

n at !t e !!

sinbt !! !cosbt !! at !e sinbt !

at !e cosbt !

}

!!F(s − a) !! sF(s)− f (0) !! 2 !s F(s)− sf (0)− f '(0) ! n (n) !!(−1) F (s) !!

1 !!! s −a n! !!! n+1 !! s n! !!! n+1 !!(s − a) b !!! s + b2 s ! 2 s + b2 b !! 2 2 !!(s − a) + b 2

s −a 2 2 !!(s − a) + b

!!

! Example!2:!!Solve! y"− 3 y'+ 2 y = tet ;

y(0)= 1,

y'(0) = 5 !!!

As!an!intermediate!step,!solve!for!Y(s),!the!Laplace!transform!of!the!solution!y(t).!! Write!your!answer!as!one!simplified!fraction.! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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Example!3:!!Solve! y"+ y = t; ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

y(0) = 1,

y '(0) = −1 !!!

! ! Example!4:!!If!I(t)!denotes!current!(in!amps)!in!the!LRC-series!electrical!circuit! shown!in!the!figure!above,!then!the!voltage!drops!across!the!inductor!(L!in!henries),! resistor!(R!in!ohms),!and!capacitor!(C!in!farads)!are!given!by!

dI q EC = , E R = RI, E L = L dt !! C where!L,!R!and!C!are!constants,!q!is!the!charge!(in!coulombs),!and!Kirchoff’s!Voltage! Law!states!that!the!impressed!voltage!E(t)!(from!a!generator!or!a!battery)!on!a! closed!loop!must!equal!the!sum!of!the!voltage!drops!in!the!loop.!!Therefore,!! ! ! ! Since!current,!I,!is!the!rate!of!change!of!charge,!q,!we!have:!! ! The!differential!equation!becomes:! ! ! !

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Use!the!Laplace!transforms!to!find!the!charge!q(t),!when!L!=!1!h,!R!=!20!ohms,!! C!=!0.005!f,!E(t)!=!150!V,!given!t!>!0,!q(0)=0!and!I(0)!=0.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 7.5!Group!Exercises! ! 1.!!Find!the!current,!I(t),!for!Example!4!above.! ! ! ! ! ! ! 2.!!For!the!IVP! y"− 4 y'+5 y = 4e3t ; y(0) = 2, y '(0) = 7 !apply!Laplace!transforms! ! to!both!sides!of!the!equation!and!solve!for!Y(s).!!Simplify!your!answer!(write!as!one! fraction,!no!complex!fractions).!!Predict!the!form!of!the!solution!without!finding!the! constant!coefficients.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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7.6(Transforms(of(Discontinuous(Functions( ( In!application!problems,!we!often!encounter!switches!that!can!be!either!off!or!on.!! The!unit(step(function,!also!call!the!Heaviside(function,!can!be!used!to!model! electrical!switches!or!valves!in!the!following!manner:!the!function!value!is!0!when! the!switch!is!off,!and!1!when!the!switch!is!turned!on!at!some!point!in!time,!t!=!a.!!We! write:! ! u(t − a) = !!! ! ! ! ! ! Sketch!a!graph.!!! ! ! Piecewise-defined!functions!can!be!written!in!compact!form!using!the!unit!step! function!and!Laplace!transforms!can!be!generalized!for!this!form.! ! Example!1:!Sketch!the!graph!of!each!function!below.! ! (a)!! ! f (t ) = (2t − 3)u(t − 3) !! ! ! ! ! ! ! ! (b) ! f (t ) = 2− 3u(t − 2) + u(t − 3) !! ! ! ! ! ! ! (c)!! f (t ) = 3t − 3tu(t − 2) !! ! ! ! ! ! ! !

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Example!2:!!Write!each!piecewise!function!using!unit!step!functions.! ! ⎧⎪ f (t ) 0 ≤ t < a (a)!! y = ⎨ !!! t ≥a ⎪⎩ g(t ) ! ! ! ! ⎧ 0 0 ≤t < a ⎪ (b)!! y(t ) = ⎨ f (t ) a ≤ t ≤ b !! ⎪ 0 t >b ⎩ ! ! Laplace!transforms!of!unit!step!functions.! ! L u(t − a) = !!

{

}

! proof:! ! ! ! ! Theorem(8.((Let! F(s) = L ! ! L f (t − a)u(t − a) = !

{

{ f (t )}(s) !!where!a!is!a!positive!constant.!

}

! proof:! ! ! ! ! ! !

{

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This!means!that! L −1 e −as F(s) = ! !! ! ! And!if!!g(t + a) = f (t ) ,!then!! ! L g(t )u(t − a) = !

{

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{

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Example!3:!!Find! L t 2u(t − 1) !!! !! ! ! ! ! ! Example!4:!!Find! L (cost )u(t − π ) !

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! ! ! ⎧ e −3s ⎫ Example!5:!!Find! L −1 ⎨ 2 ⎬ !!! ⎪⎩ s +16 ⎪⎭ ! ! ! ! ! Example!6:!!Solve!the!introductory!mixing!problem!from!Section!7.1.!!Recall!that!we! had!modeled!the!problem!with!the!following!IVP:! ! ! ! ! ! ! ! (1)!write!the!forcing!function!using!unit!step!functions! (2)!find!the!Laplace!transform!of!both!sides! (3)!solve!for!X(s)! (4)!apply!inverse!transforms!to!solve!for!x(t).! ! ! ! ! ! ! ! ! ! ! ! ! !

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A(Brief(Table(of(Laplace(Transforms( ( F(s) = L f (t ) (s) !!! ! f (t ) !!

{

at !e f (t ) ! f '(t )! f "(t ) ! n !t f (t )! at !e !!

n !t !!

n at !t e !!

sinbt !! !cosbt !! at !e sinbt !

at !e cosbt !

! f (t − a)u(t − a), a > 0 !! g(t )u(t − a), a > 0 ! !!u(t − a), a > 0 !

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!!F(s − a) !! sF(s)− f (0) !! 2 !s F(s)− sf (0)− f '(0) ! n (n) !!(−1) F (s) !!

1 !!! s −a n! !!! n+1 !! s n! !!! n+1 !!(s − a) b !!! s + b2 s ! 2 s + b2 b !! 2 2 !!(s − a) + b 2

s −a 2 2 !!(s − a) + b −as !!e F(s) !!!

{

!!

}

e −as L g(t + a) (s) !!!

e −as !!! ! s

! 7.6!Group!Exercises! ! 1.!!For!the!graph!given!in!the!margin,!write!the!function! f(t)!using!unit!step!functions.!!Then!find!the!Laplace! transform!of!f(t).! ! ! ! ! ! ⎧ e −3s ⎫ 2.!!Find! L −1 ⎨ 3 ⎬ !! ⎩⎪ s ⎭⎪

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3.!!Express!f!in!terms!of!unit!step!functions.!

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! Find!the!Laplace!transform!of!f(t).! ! ! ! ! ! ⎧ e− s ⎫ −1 L 4.!!Find! ⎬! ⎨ ⎪⎩ s(s +1) ⎪⎭ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ( (

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7.8(Convolution( ( Suppose!we!wished!to!solve!the!initial!value!problem,!where!g(t)!is!an!unknown! function:! y"+ y = g(t ); y(0) = 0, y '(0) = 0. !!We!would!like!to!express!the!solution! ! in!terms!of!the!unknown!function!g(t):! ! ! ! ! ! ! We!can!see!that!the!solution,!y(t),!will!involve!some!relationship!between!sin!t!and! g(t);!!we!will!call!it!the!“convolution”!of!sin!t!and!g(t),!and!will!use!the!notation!! sin!t!*!g(t).! ! It!can!be!shown!that!the!following!definition!of!convolution!will!result!in!the! required!Laplace!transform:! ! Definition:!Let!f(t)!and!g(t)!be!piecewise!continuous!functions!on! [0, ∞) !.!!The! convolution!of!f(t)!and!g(t)!is!given!by! ! (f!*!g)(t)!=!! ! It!can!easily!be!shown!that!f!*!g!=!g!*!f.! ! Example!1:!!Find!each!convolution:! ! (a)!! !t! *t 2 !! ! ! ! (b)!! 1*t !! ! ! ! ! Theorem(11(Convolution(Theorem( ( Let!f(t)!and!g(t)!be!piecewise!continuous!functions!and!of!exponential!order!on! [0, ∞) .!!Then!! L { f * g} = L { f (t )}L { g(t )} = F(s)G(s) !.! ! ! !

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Example!2:!!Find!! ! (a) !!L {t *t 2 } ! ! (b)!!! L

{∫

t

0

}

eν sin(t − ν ) d ν !!

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⎧ 1 ⎫ (c)!! L −1 ⎨ 2 !! 2⎬ ⎩(s +1) ⎭ !! ! ! ! ! Example!3:!!Use!convolution!to!complete!the!solution!the!initial!value!problem!in! terms!of!g(t):!!! y"+ y = g(t ); y(0) = 0, y '(0) = 0. ! ! ! ! ! ! ! ! Example!4:!!Find!! ! (a)!! !1* f (t )!=! ! (b)!! L 1* f (t ) = ! ! ! ⎫ −1 ⎧ 1 (c)!!therefore,! L ⎨ F(s)⎬ = !!! ⎭ ⎩s ! ! ! Example!5:!Use!the!result!of!Ex.4(c)!to!find!each:! ! ⎫ ⎧ 1 (a)!! L −1 ⎨ 2 ⎬ = !!! ⎩s(s +1) ⎭

{

}

! (b)!! L !! ! !

−1

⎫ ⎧ 1 ⎬= ! ⎨ 2 2 ⎩s ( s +1) ⎭

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The(Transfer(and(Impulse(Response(Functions( ( The!transfer(function!H(s)!of!a!linear!differential!equation!system!is!defined!as!the! ratio!of!the!Laplace!transform!of!the!output!function!y(t)!to!the!Laplace!transform!of! the!input!function!g(t),!under!the!assumption!that!all!initial!conditions!are!zero.!! That!is,! ! H(s)!=!! ! ! If!the!linear!system!is! !ay"+ by '+ cy = g(t ), t > 0 !,!then!take!Laplace!transforms!of! both!sides!to!get! ! ! ! ! ! And!so!H(s)!=! ! ! ! The!impulse(response(function!for!the!system!is!h(t)!=! ! Physically,!it!describes!the!solution!when!a!mass-spring!system!is!struck!by!a! hammer!(a!homogeneous!problem!with!initial!values!h(0)!=!0!and!h’(0)!=!1/a).! ! The!impulse!response!function!can!be!used!to!find!the!solution!to!the!general!initial! value!problem!using!the!Impulse(Response(Theorem:! ! Let!I!be!an!interval!containing!the!origin.!!The!unique!solution!to!the!initial!value! problem! ! ay"+ by '+ cy = g(t ), y(0) = y0 , y'(0)= y1 !! ! where!a,!b!and!c!are!constants!and!g!is!continuous!on!I,!is!given!by! ! y(t) = (h * g)(t) + yk (t) = ! ! where!h!is!the!impulse!response!function!for!the!system!and!yk!is!the!unique!solution!to!the! homogeneous!IVP:! !

ay"+ by'+ cy = 0, y(0) = y0 ,

y '(0) = y1 !

! !

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Example!1:!A!linear!system!is!governed!by!the!differential!equation!! y"+ 9 y = g(t ); y(0) = 2, y'(0) = −3 .! ! ! Find!the!transfer!function,!H(s)!=! ! ! Find!the!impulse!response!function!h(t)!=! ! ! Find!the!solution!to!the!homogeneous!IVP,!!!yk!=! ! ! ! ! ! ! ! Use!convolution!to!write!the!solution!to!the!general!differential!equation:! ! y(t)!=!! ! ! ! ! Example!2:!A!linear!system!is!governed!by!the!differential!equation!! y"− 9 y = g(t ); y(0) = 2, y'(0) = 0 .! ! ! Find!the!transfer!function,!H(s)!=! ! ! Find!the!impulse!response!function!h(t)!=! ! ! Find!the!solution!to!the!homogeneous!IVP,!!!yk!=! ! ! ! ! ! Use!convolution!to!write!the!solution!to!the!general!differential!equation:! ! y(t)!=!! ! !

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Section!7.8!Group!Exercises! ! ⎧ 1 ⎫ 1.!!Find! L −1 ⎨ 2 ⎬ !!three!different!ways,!and!get!three!“different”!answers:! ⎩ s −1 ⎭ ! ! (a)!use!a!table!entry! ! ! ! ! (b)!use!partial!fractions! ! ! ! ! (c)!use!convolution! ! ! ! ! ! 2.!!A!linear!system!is!governed!by!the!differential!equation! y"+ 2 y '+ 5y = g(t); y(0) = 2, y '(0) = −2 !.! Find!the!transfer!function,!the!impulse!response!function,!and!a!formula!for!the! solution.! ! ! !

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