18.03SC Unit 3 Practice Exam and Solutions
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1 Sudy Guide on Sep, Dela, Convoluion, Laplace You can hink of he ep funcion u() a any nice mooh funcion which i for < a and for > a, where a i a poiive number which i much maller han any ime cale we care abou in he conex we are udying a he momen. Similarly, he be way for you o underand he dela funcion i o hink of i a any mooh funcion which i zero excep in he immediae neighborhood of = and which ha inegral.. So u( b) = δ( b) and if a < b hen δ(τ b) dτ = u( b). a A funcion f () i regular or piecewie mooh if i can be broken ino piece each having all higher derivaive and uch ha a each breakpoin f (n) (a ) and f (n) (a+) exi. A ingulariy funcion i a linear combinaion of hifed dela funcion. A generalized funcion f () i a um f () = f r () + f () of a regular funcion and a ingulariy funcion. Any regular funcion f () ha a generalized derivaive f ' (), wih regular par f ' () he r regular derivaive of f () whereever i exi, and ingular par f ' () given by a um of erm ( f (a+) f (a ))δ( a), one for each break in he graph of f (). c c+ Then f ' () d = f (c) f (a). (To be more precie, f ' () d = f (c+) f (a ).) a a For he re of hi uni, all ignal (funcion of ) are uppoed o be zero for <. We look for oluion o he differenial equaion p(d)x = q(), epecially wih re iniial condiion, o ha x() = for < and x ha a many derivaive a poible. The uni impule repone or weigh funcion of he operaor p(d) i he oluion w() o he equaion p(d)w = δ(), wih re iniial condiion. If p() = a n n + + a wih a n =, and x i uch ha p(d)x = and x() = = x (n ) () = and x (n ) () = a n, hen w() = u()x(). The uni ep repone i he oluion o p(d)v = u() wih re iniial condiion; for > hi coincide wih he oluion o p(d)x = uch ha x() = =.. x (n ) () =. Since u() = δ(), v() = w(). The repone o a uni impule deermine he yem repone o any oher inpu ignal (wih re iniial condiion): he oluion o p(d)x = q() i given by x() = w() q(), where he aerik indicae he convoluion produc f () g() = f ( τ)g(τ) dτ. The convoluion produc ha properie analogou o he ordinary produc: f (g h) = ( f g) h, f (ag + bh) = a( f g) + b( f h), f g = g f. Alo f () δ() = f (). If you feed he oupu of a yem (wih uni impule repone g()) ino anoher yem (wih uni impule repone f ()), you ge a compoie yem wih uni impule repone f () g(). The Laplace ranform carrie a generalized funcion (wih f () = for < ) o a funcion F() of a complex variable. I obey a bunch of rule, he mo imporan of which are lineariy and he -derivaive rule L[ f ' ()] = F(), where here f ' () denoe he generalized derivaive. There are andard compuaion, oo, including L[δ()] = (he conan funcion of ). Thee already imply ha he uni impule repone w() of p(d) aifie L[w()] =. W() = L[w()] i he ranfer funcion of he operaor p(d). Alo he olup() ion o p(d)x = f () wih re iniial condiion ha Laplace ranform X() = W()F().
2 OCW 8.3SC Thi relae o he formula L[ f () g()] = F()G(). If f () and g() have he ame Laplace ranform, hen f (a ) = g(a ) and f (a+) = g(a+) for every a. If F() i a raional funcion (quoien of a polynomial by a polynomial), coninued fracion provide andard way o find f (). Coverup i an efficien way o compue ome of he coefficien. Thi couple wih compleing he quare and he -hif rule. The pole diagram of F() i he e of complex number a which F() become infinie. Thi i he e of uncancelled zero in i denominaor. The pole diagram of F() = L[ f ()] deermine he region of convergence of he inegral: i i he region o he righ of he verical line hrough he righmo pole. The pole diagram alo conrol much of he behavior of f () for large (while aying nohing abou behavior for mall ). The righmo pole dominae. A pole a a + iω lead o exponenial growh/decay like e a (or ome polynomial ime e a ) and ocillaion of circular frequency ω. The ranfer funcion occured earlier in he coure: he exponenial oluion o p(d)z = e r i z p = W(r)e r. So he inuoidal oluion o p(d)x = co(ω) i x p = W(iω) co(ω φ), where φ = Arg (W(iω)): W(iω) i he complex gain if e r ielf i regarded a he inpu ignal), and W(iω) i he gain. The ampliude repone curve i obained by inerecing he graph of W() wih he plane above he imaginary axi. Pracice Hour Exam. Le ω be a poiive conan. We drive a harmonic ocillaor wih a quare wave of.. circular frequency ω: x + 4x = q(ω). (a) Wrie down a periodic oluion o he equaion, if ω i uch ha here i one. (b) For wha value of ω doe here fail o be a periodic oluion? for <. Le f () = for < <. for > (a) Skech he graph of f (). (b) Skech he graph of he generalized derivaive f ' (). (c) Wrie down a formula for f ' () in erm of ep and dela funcion. 3. (a) Compue he convoluion produc 6. (b) A cerain operaor p(d) ha uni impule repone w() = u()e. Wha i he oluion o p(d)x = e wih re iniial condiion? (a) Wha i he Laplace ranform of he oluion o he equaion x + x + x = having re iniial condiion? (b) Wha funcion f () ha Laplace ranform F() =? ( + )( + + 5) 5. (a) Skech he pole diagram for he funcion F() =. ( + )( + + 5) (b) Give an example of a funcion f () whoe Laplace ranform ha pole a = and = 3 ± 4i and nowhere ele.
3 OCW 8.3SC Properie of he Laplace ranform. Definiion: L[ f ()] = F() = f ()e d for Re >>.. Lineariy: L[a f () + bg()] = af() + bg().. Invere ranform: F() eenially deermine f (). 3. -hif rule: L[e a f ()] = F( a). f ( a) if > a 4. -hif rule: L[ f a ()] = e a F(), f a () =. if < a 5. -derivaive rule: L[ f ()] = F ' (). 6. -derivaive rule: L[ f ' ()] = F(), where f ' () denoe he generalized derivaive. ' L[ f ()] = F() f (+) if f () i coninuou for >. r 7. Convoluion rule: L[ f () g()] = F()G(), f () g() = f ( τ)g(τ)d τ. 8. Weigh funcion: L[w()] = W() = /p(), w() he uni impule repone. Formula for he Laplace ranform n! L[] = L[e a ] = L[ n ] = a n+ ω L[co(ω)] = L[in(ω)] = + ω + ω ω ω L[ co(ω)] = L[ in(ω)] = ( + ω ) ( + ω ) Fourier coefficien for periodic funcion of period π: a f () = + a co() + a co() + + b in() + b in() + π π a m = f () co(m) d, b m = f () in(m) d π π π π If q() i he odd funcion of period π which ha value beween and π, hen 4 in(3) in(5) q() = in() π 3 5 3
4 OCW 8.3SC Soluion ( ) 4 in(3ω) in(5ω). (a) q(ω) = in(ω) Uing uperpoiion and he π in(ω) fac ha x + 4x = A in(ω) ha oluion x p = A 4 ω, ( ) 4 in(ω) in(3ω) in(5ω) x p = + + π 4 ω 4 9ω 4 5ω + (b) Thi oluion doe no exi if ω = /(odd ineger).. (a) or. (b) or (c) f ' () = (u() u( )) + δ() + δ( ). ( ) τ 7 τ (a) 6 = ( τ)τ 6 dτ = (τ 6 τ 7 ) dτ = = 8 = ( τ) (b) x() = w() e = ( τ)e e τ dτ = e ( τ) dτ = e ( τ) = e 4. (a) X + X + X =, o X =. ( + + ) a b( + ) + c (b) = +. ( + )( + + 5) + ( + ) + 4 ( ) ( + i) By coverup, a = =, b(i) + c = = + i o b = and ( ) + ( ) + 5 i c =. Thu f () = e + e co() + in(). 5. (a) Pole a = and a = ± i. (b) e + e 3 in(4), or many oher. 4
5 MIT OpenCoureWare hp://ocw.mi.edu 8.3SC Differenial Equaion Fall For informaion abou ciing hee maerial or our Term of Ue, vii: hp://ocw.mi.edu/erm.
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