Chapter 6. Systems of First Order Linear Differential Equations

 Juliet Higgins
 10 months ago
 Views:
Transcription
1 Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh change in our noaion for DE s Before, in Chapers 4, we used he leer x for he independen variable, and y for he dependen variable For example, y = sinx, or x dy + xy = sinx Now we will use for he independen dx variable, and x,y,z, or x,x,x 3,x 4, and so on, for he dependen variables For example: x = sin x = cos And when we wrie x, for example, we will henceforh mean dx d The firs order sysems (of ODE s) ha we shall be looking a are sysems of equaions of he form x = expression in x,x, x n, x = expression in x,x, x n, x n = expression in x,x, x n,, valid for in an inerval I These expressions on he righ sides conain no derivaives A firs order IVP sysem would be he same, bu now we also have iniial condiions x (a) = c,x (a) = c,,x n (a) = c n Here a is a fixed number in I, and c,c,,c n are fixed consans Example x = y y = x+ (Where is?) Example Example 3 x = x x x 3 sin()x 3 x = 3x x 3 + x 3 = ex x = y y = 3 x+ y 9, x() = 3,y() = 6 These are called firs order sysems, because he highes derivaive is a firs derivaive Example 3 is a firs order IVP sysem, he iniial condiions are x() = 3,y() = 6 A soluion o such a sysem, is several funcions x = f (),x = f (),,x n = f n () which saisfy all he equaions in he sysem simulaneously A soluion o a firs order IVP sysem also has o saisfy he iniial condiions
2 For example, a soluion o Ex above is x = + sin,y = cos To check his, noice ha if x = +sin and y = cos, hen clearly x = (+sin) = cos = y, and y = sin = (+sin)+ = x+ So boh equaions are saisfied simulaneously Similarly, a soluion o he firs order IVP sysem in Ex 3 above is x = 3,y = 6 (Check i) Jus as in Chaper, under a mild condiion here always exis soluions o a firs order IVP sysem, and he soluion will be unique, bu local (ha is, i may only exis in a small inerval surrounding a) The proof is almos idenical o he one in Chaper Trick o change higher order ODE s (or sysems) ino firs order sysems: For example consider he ODE y sin(x)y +y xy = cosx Le = x,x = y,x = y,x 3 = y Then y = x 3 We do no inroduce a variable for he highes derivaive We hen obain he following firs order sysem: x = x x = x 3 x 3 = cos+x x +sin()x 3 Sraegy: solve he laer sysem; and if x = f() hen he soluion o he original ODE is y = f(x) So for example if x = 3cos() hen y = 3cos(x) Similarly a higher order IVP like y sin(x)y + y xy = cosx, y() =,y () =,y () = 3, is changed ino a s order IVP sysem (he one in he las paragraph), wih iniial condiions x () =,x () =,x 3 () = 3 Using he same rick, any nh order sysem may be changed ino a firs order sysem Combining he exisence and uniqueness resul a few bulles above, wih he rick jus discussed, we see ha every nh order IVP has a unique local soluion under a mild condiion Linear sysems A firs order linear sysem is a firs order sysem of form x = a ()x +a ()x + +a n ()x n +b () x = a ()x +a ()x + +a n ()x n +b () x n = a n ()x +a n ()x + +a nn ()x n +b n () Examples like or x = y y = 3 x+ xy 9, x = y y = 3 x + y 9,
3 are no linear (on he righ sides he dependen variables, in his case x and y are only allowed o be muliplied by consans or funcions of We will see some more examples momenarily Marix formulaion of linear sysems The coefficien marix of he las sysem is A() = a ij () Tha is a () a () a n () b () a () a () a n () b () A() =, b() = a n () a n () a nn () b n () hen he sysem may be rewrien as a single marix equaion x = A() x+ b(), x = x x x n, x = x Example Consider he sysem = x e x + x = x The coefficien marix of +cos()x he las sysem is e A() = And b() = cos If we wrie x x for, and x x for, hen he sysem may be rewrien as a single marix equaion x x = x e cos x + x x x n Example The IVP sysem in Example 3 above may be rewrien as a single marix equaion 3 x = x +, x() = Thus a firs order linear sysem is one ha can be wrien in he form x = A() x + b() Here A() is a marix whose enries depend only on, and b() is a column vecor whose enries depend only on Linear firs order IVP sysems always have (unique) soluions if A() and b() are coninuous; in fac we will give formulae laer for he soluion in he consan coefficien case (ha is when A() is consan, does no depend on ) Vecor funcions: The vecor x above depends on Thus i is a vecor funcion Similarly, b() above is a vecor funcion We call i an ncomponen vecor funcion if i has n enries, ha is if i lives in R n
4 You should hink of a soluion o he marix DE in he Definiion above as a vecor x = 3 funcion For example, you can check ha is a soluion o Example above y = 6 (which was Example 3 before) Wewriehis soluionashe vecor funcion 3 u() = 6 One can check ha indeed u = u + 9 Do i! (We did i in class) 3 The sysem above is called homogeneous if b() = If x = A() x + b(), (N) is no homogeneous hen he associaed homogeneous equaion or reduced equaion is he equaion x = A() x We can rewrie (N) as x A() x = b(), or where D x = x, or simply as (D A()) x = b() L x = b() (N) where L = D A() I is easy o see as before ha L = D A() is linear, ha is: (N) L(c u +c u ) = c L u +c L u Thus he main resuls in Chapers 3 and 5 carry over o give varians valid for firs order linear sysems, wih essenially he same proofs We sae some of hese resuls below Firs we discuss homogeneous firs order linear sysems 6 Homogeneous firs order sysems Here we are looking a x = A() x, (H) for in an inerval I Thus is jus L x = where L = D A() as above We will fix a number n hroughou his secion and he nex, andassume ha we have n variables x,,x n, each a funcion of So A() is an n n marix We will hen refer o (H) someimes as (H) n, reminding us of his fixed number n, so ha eg A() is n n, ec The proofs of he nex several resuls are similar (usually almos idenical) o he maching proofs in Chaper 3 (and Chapers 5 and ) Of course he zero vecor is a soluion of (H) As before, his soluion is called he rivial soluion
5 Theorem If u and u are soluions o (H) on I hen so is u + u and c u (x) soluions o (H), for any consan c So he sum of any wo soluions of (H) is also a soluion of (H) Also, any consan muliple of a soluion of (H) is also a soluion of (H) Again, a linear combinaion of u, u,, u n is an expression for consans c,,c n c u +c u + +c n u n, The rivial linear combinaion is he one where all he consans c k are zero This of course is zero Theorem Any linear combinaion of soluions o (H) is also a soluion of (H) Two vecor funcions u and v whose domain includes he inerval I, are said o be linearly dependen on I if u is a consan imes v, or v is a consan imes u If hey are no linearly dependen hey are called linearly independen Anoher way o say i: u and v are linearly independen if he only linear combinaion of u and v which equals zero, is he rivial one More generally, u, u,, u k are linearly independen if no one of u, u,, u k is a linear combinaionofheohers, noincluding iself Equivalenly: u, u,, u k arelinearly independen if he only way c u ()+c u ()+ +c k u k () = for all in I, for consans c,,c n, is when all of hese consans c,,c n are zero TheWronskianofnncomponenvecorfuncions u, u,, u n,wrienw( u, u,, u n )() or W() or W( u, u,, u n ), is he deerminan of he marix u : u : : u n This las marix is he marix whose jh column is u j () Proposiion If u, u,, u n are linearly dependen on an inerval I, hen for all in I W( u, u,, u n )() = Proof This follows from he equivalence of () and (8) in he par heorem proved in Homework Corollary If W( u, u,, u n )( ) a some poin in I hen u, u,, u n are linearly independen Theorem There exis n soluions u, u,, u n o (H) n which are linearly independen
6 Proof Similar o he maching proof in Chaper 3, or we will give a formula for he soluion laer n soluions u, u,, u n o (H) n which are linearly independen, are called a fundamenal se of soluions o (H) Then he marix X() = u : u : : u n me above is called he fundamenal marix Theorem If u, u,, u n are soluions o (H) n on an open inerval I hen eiher W( u, u,, u n )() = for all in I; or W( u, u,, u n )() for all in I This means ha an n n marix X() whose columns are soluions o (H) n is a fundamenal marixif andonlyif X()is inverible forall ini, andif andonlyif X()is inverible for some in I By he par heorem proved in Homework his can be phrased in many equivalen ways If u, u,, u n are soluions o (H) n on I, and if every soluion o (H) n on I is of he form c u +c u + +c n u n, for consans c,,c n, hen we say ha c u +c u + +c n u n is he general soluion o (H) n Theorem Suppose ha u, u,, u n are soluions o (H) n on an open inerval I The following are equivalen: (i) u, u,, u n are a fundamenal se of soluions o (H) on I, (ii) W( u, u,, u n )() for some (or all) in I, (iii) c u +c u + +c n u n is he general soluion o (H) Example Show ha u = sysem 3 3 and v = x = y y = 3 x+ y are a fundamenal se for he linear on he inerval (, ) Also, find he general soluion o his sysem Soluion This is he sysem x = A() x where A() = 3 Check ha u = A() u (we checked his in class), and v = A() v Then noe ha 3 W( u, v) = de( 3 ) = 3 ( ) 3 = 3 = 4 So by he las heorem u, v is a fundamenal se of soluions, and he general soluion o his sysem is C u + D v Tha is, he general soluion is x = C 3 + D and y = 3C D (Explained in more deail in class)
7 63/64 Homogeneous firs order sysems wih consan coefficiens III Here we are looking a x = A x, (H) for in an inerval I Here A is an n n marix wih consans (numbers) as enries Again we are fixing a number n hroughou, and assume ha we have n variables x,,x n, each a funcion of We will again refer o (H) someimes as (H) n, reminding us of his fixed number n, so ha eg A is n n, ec If λ is an eigenvalue of A wih eigenvecor v, se x = e λ v Noe ha x = e λ λ v, whereas A x = e λ A v = e λ λ v So x = A x; ha is e λ v is a soluion o (H) Firssupposehahen nmarixahasnlinearlyindependeneigenvecors v, v,, v n, and ha λ k is he eigenvalue associaed wih he eigenvecor v k Then he general soluion is x = c e λ v +c e λ v + c n e λn v n, where c,c,, are arbirary consans Tha is, a fundamenal se is: e λ v,e λ v,,e λn v n The fundamenal marix X() is he marix wih hese as columns Proof: The Wronskian W() = de(x()) by () (8) in he par heorem in Homework, since X() is he marix wih columns v, v,, v n, which are linearly independen Ofcourse, ifhen nmarixahasndisinc eigenvalues, henifwe findoneeigenvecor for each eigenvalue, hese will be linearly independen by Theorem in Secion 58 Also, from more advanced linear algebra i is known ha if A = A T, ha is if A is symmeric, hen here will exis n linearly independen eigenvecors So he mehod above will work x = x 3y Example Solve y = x+y Soluion This is jus x = A 3 x, where A = We firs find he eigenvalues and corresponding eigenvecors of A We have λ 3 de(a λi) = λ = ( λ)( λ) 6 = λ 3λ 4 = (λ+)(λ 4) Thus he eigenvalues are λ = and 4 To find an eigenvecor corresponding o λ = 4 we need o solve A x = 4 x We solve 3 = 4??
8 Fromhefirs row, weseeha 3? = 4, soha? = Thus aneigenvecor corresponding o λ = 4 is Similarly, o find an eigenvecor corresponding o λ = we solve 3 =?? Thus 3? = so ha? = Thus an eigenvecor corresponding o λ = is:, 3 /3 3 or, muliplying by 3, we ge he eigenvecor A general soluion o he sysem is hen 3 x = c e 4 +c e, where c,c are arbirary consans We can rewrie his as c e x = 4 +3c e c e 4 +c e This corresponds o x = c e 4 +3c e he soluion y = c e 4 +c e x = x +x x 3 Example Find he general soluion o x = x +3x x 3 Also solve he IVP x 3 = x +x +x 3 consising of his sysem wih iniial condiions x () =,x () =,x 3 () = Soluion This is jus x = A x, where A = 3 One can show ha his marix A has eigenvalues,, (I will skip he work, i is jus as in he previous chaper) For he eigenvalue we only wan one eigenvecor, and you can check ha is an eigenvecor (using he usual mehod as in he previous chaper, similar o wha follows:) To find evecors for he eigenvalue we mus solve (A I) x =, which is x = This has soluion (afer Gauss eliminaion): x = s x = s x 3 = Seing s =, =, hen =,s =, we ge linearly independen eigenvecors ;
9 Thus he general soluion o he sysem is x = c e + c e + c 3 x = c e +c e c 3 e Reading his row by row, we can rewrie his as x = c e +c e x 3 = c e +c 3 e e To solve he IVP we mus solve c e +c e +c 3 e = Tha is, c +c +c 3 = Using he soluion mehod of your choice (row reducion, inverse, Cramers rule), he soluion is: c =,c =,c 3 = The soluion of he iniialvalue problem is x = e + e + c c c 3 = e Reading his row by row, we can rewrie his as x = e x = e +e x 3 = e +e Wha if some of he eigenvalues are complex? If A is a real marix, hen is complex eigenvalues occur in pairs α±iβ, wih α,β real Suppose ha v is an eigenvecor corresponding o an eigenvalue λ = α+iβ Wrie v = r +i s where r and s have only real enries Then in he general soluion o he linear sysem include erms: Ce α ( r cos(β) s sin(β)) + De α ( r sin(β)+ s cos(β)) Here C,D are consans (This also akes care of he α iβ case, so ignore ha case) Example 3 Solve he linear sysem x = 3 3 λ Soluion We have de(a λi) = λ = (3+λ)(+λ)+ = λ +4λ+5 Using he quadraic equaion b± b 4ac you can easily show ha λ = ± i are he a wo eigenvalues To find an eigenvecor corresponding o λ = + i we need o solve A x = ( +i) x So we solve 3 = ( +i)?? x
10 From he firs row, we see ha 3? = +i, so ha? = i Thus an eigenvecor is We wrie his as r +i s as above: i = + i i Thus, by he discussion above he Example, he general soluion is ( ) ( x = Ae cos sin + Be sin + where A,B are arbirary consans This may be rewrien Reading row by row we ge: x = Ae cos+be sin x = Ae ( cos+sin)+be ( sin cos) cos ), Wha if A is n n, bu you canno find n linearly independen eigenvecors? In his case, if λ is an eigenvalue of A of mulipliciy k, bu here are fewer han k linearly independen eigenvecors for A, hen use generalized eigenvecors for λ For example, if λ is an eigenvalue of mulipliciy, bu i has a mos one linearly independen eigenvecor v, solve he equaion (A λi) w = v Then a linearly independen pair of soluion vecors corresponding o λ are e λ v (as before) and e λ w +e λ v So as par of he general soluion o (H) we will have Ce λ v +De λ ( w + v) If λ is an eigenvalue of mulipliciy 3 and one can only find wo linearly independen eigenvecors v and z, hen a hird soluion vecor corresponding o λ is e λ w+e λ v where w is a soluion o he equaion (A λi) w = v So as par of he general soluion o (H) we will have Ce λ v +De λ z +Ee λ ( z + v) If λ is an eigenvalue of mulipliciy 3, bu i has a mos one linearly independen eigenvecor v, solve he equaions (A λi) w = v, (A λi) z = w Then hree linearly independen soluion vecors corresponding o λ are e λ v (as before) and e λ w +e λ v, and e λ z +e λ w + e λ v So as par of he general soluion o (H) we will have Ce λ v +De λ ( w + v)+ee λ ( z + w + v) Examples worked in class (p 98 3 in Tex) (a) Find he general soluion o he sysem x = x 3
11 (b) Find he general soluion o he sysem x = Soluion (a) A = of A We have de(a λi) = 3 λ 3 λ 3 x We firs find he eigenvalues and corresponding eigenvecors = ( λ)(3 λ)+ = λ 4λ+4 = (λ ) Thus he eigenvalues are λ = (mulipliciy ) To find an eigenvecor corresponding o λ = we need o solve A x = x We solve (A I) x = The usual Gauss eliminaion (done in class) yields x =, x = so we are only able o find a mos one linearly independen eigenvecor v = The recipe above now ells us o solve (A I) x = v, ha is, x = The usual Gauss eliminaion (done in class) yields a soluion o his of x = s x = s We can ake s o be anyhing, say s =, giving a soluion vecor w = A general soluion o he sysem is hen Ce λ v +De λ ( w + v), or x = Ce +De ( + ),, where c,c are arbirary consans We can rewrie his as x = Ce +De ( ) x = Ce +De (b) Is very similar o (a) and is solved on page 99 of he online exbook Or a leas one can find here, in he second las line of ha soluion, a fundamenal se of hree soluions x, x, x 3 The general soluion is x = c x +c x +c 3 x 3
12 There is anoher way o solve any linear sysem x = A x, (H) wihou using eigenvalues and eigenvecors a all! Bu we will need Marix exponenials: if A is an n n marix consider he sum I +A+ A + 3! A3 + 4! A4 + This is he power series formula for e A from Calculus II I is no hard o show ha his converges in he sense of Calculus o an n n marix which we wrie as exp(a) Or, simply ge a compuer o add he firs 5 erms of his sum o approximae i wih grea precision Theorem Le Ψ() = exp(a) Then Ψ() is a fundamenal marix for (H) The n columns of his marix are a fundamenal se for (H) (Noe ha he jh column is Ψ() e j ) So he general soluion o (H) is x = Ψ() Indeed, for any vecor a, a soluion o he IVP sysem which is (H) ogeher wih iniial condiion x() = a, is Ψ() a Proof One can show easily ha d (exp(a)) = Aexp(A) = exp(a)a Thus if a is any d vecor, and if v = Ψ() a hen c c c n d d v = d (exp(a)) a = Aexp(A) a = A v d Thus v = Ψ() a is a soluion o he linear sysem x = A x Seing v = e j, we see ha he columns of Ψ() are soluions o (H) As we said earlier, o check Ψ() is a fundamenal marix for (H) we need only check ha Ψ() is inverible However Ψ() = Ψ( ) since exp(a) exp( A) = exp(a A) = exp() = I Since Ψ() = I, if v = Ψ() a hen v() = I a = a Thus v is a soluion o he linear IVP sysem x = A x, x() = a Similarly, a soluion o he linear IVP sysem x = A x, x( ) = a is exp(( )A) a = exp(a)exp( A) a 65 Solving nonhomogeneous sysems Here we are looking a x = A x + b(), (N)
13 for in an inerval I Here A is an n n marix wih consans (numbers) as enries Again we are fixing a number n hroughou, and assume ha we have n variables x,,x n, each a funcion of The reduced equaion or associaed homogeneous equaion for (N) is x = A x, (H) Key fac: As in Chapers 3 (and 5 and ), finding he general soluion o x = A() x + b(), breaks ino wo seps: Sep : find he general soluion o he associaed homogeneous equaion (H) Sep : Find one soluion (called a paricular soluion) o (N) Then add wha you ge in Seps and (The proof of his is he same as before) The proof is he same as we saw in Chapers 3 (and 5 and ): le L x = x A x This is linear So if x p is a paricular soluion o (N) and x H is a soluion o (H), hen L( x p + x H ) = L x p +L x H = b+ = b If z is any soluion o (N) hen z x p is a soluion o (H), since L( z x p ) = = So if x H is his soluion o (H), z = x p + x H The basic formula in his secion is ha a paricular soluion o he linear sysem x = A() x + b() is given by he formula x = X() X(s) b(s)ds, where X() is he fundamenal marix This is variaion of parameers, and a proof can be found on page 35 of he ex book Les see wha his means, and how o use i in examples: Example Find he general soluion o he linear sysem x 4 3 A = and b() = e = A x + b(), where Noe ha his is he same as asking you o find he general soluion o he linear sysem x = 4x 3x + x = x You need o be able o jump beween boh ways of wriing he x +e sysem Soluion: Firs we solve he associaedhomogeneous sysem x = A x, using hemehods of 63 I will skip he working (see 63), and jus wrie down he answer: 3 x = c e + c e e 3e Thus he fundamenal marix is X() = e e (see he discussion in Secion 6 above) To find a paricular soluion o he original sysem, by he basic formula above, we need o compue X() X(s) b(s)ds
14 We have X(s) = Hence Now inegrae his: e s 3e s e s e s X(s) = e 3s X(s) b(s) = The deerminan of his is e 3s 3e 3s = e 3s Thus e s 3e s e s e s e s 3e s e s X(s) b(s)ds = e s = s e s e s 3e s e s = ( se s +3)ds (se s e s )ds e s se s +3 se s e s Now ( se s +3)ds = e (+)+3 ; and (se s e s )ds = e ( 4 )+e 3 4 (Check  his jus uses Calculus inegrals) Thus X(s) b(s)ds = paricular soluion e 3e e e e (+)+3 e ( 4 )+e 3 4 e (+)+3 e ( 4 )+e 3 4 = Muliplying his by X() we ge he (3+)e 9 4 e e 3 e The las sep here looks complicaed, bu is jus algebra (Check i!) On he es, he algebra won work ou quie as ugly Thus by he Key Fac (menioned a he sar of he secion above), he general soluion o our original sysem is + 5 x = 4 +(3+)e 9 4 e e 3 + c e + c e e The general soluion o equaion (N) is c c x = X() +X() X(s) b(s)ds, a c n since he general soluion o he reduced equaion (H) is c c X() c n I follows ha he soluion o he IVP which is equaion (N) wih iniial condiion x(a) = x is x = X() X(s) b(s)ds+x()x(a) x, a where X() is he fundamenal marix
15 Proof: seing = a in he las formula we ge x(a) = X(a) a a X(s) b(s)ds+x(a)x(a) x = +I x = x So he iniial condiion is saisfied Also, ha las formula equals where X() X(s) b(s)ds+x() w +X() y = X() w = a X(s) b(s)ds+x() c, X(s) b(s)ds, y = X(a) x, c = w + y Bu X() X(s) b(s)ds+x() c is of he form of he general soluion o (N) in he second las bulle
ODEs II, Lecture 1: Homogeneous Linear Systems  I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems  I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationt + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that
ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he secondorder ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so
More informationAfter the completion of this section the student. Theory of Linear Systems of ODEs. Autonomous Systems. Review Questions and Exercises
Chaper V ODE V.5 Sysems of Ordinary Differenial Equaions 45 V.5 SYSTEMS OF FIRST ORDER LINEAR ODEs Objecives: Afer he compleion of his secion he suden  should recall he definiion of a sysem of linear
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More informationKEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More informationHOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.
HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More information8. Basic RL and RC Circuits
8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 51 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationUndetermined coefficients for local fractional differential equations
Available online a www.isrpublicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
More informationKEY. Math 334 Midterm III Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Fall 28 secions and 3 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationIntroduction to Probability and Statistics Slides 4 Chapter 4
Inroducion o Probabiliy and Saisics Slides 4 Chaper 4 Ammar M. Sarhan, asarhan@mahsa.dal.ca Deparmen of Mahemaics and Saisics, Dalhousie Universiy Fall Semeser 8 Dr. Ammar Sarhan Chaper 4 Coninuous Random
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationChapter 7 Response of Firstorder RL and RC Circuits
Chaper 7 Response of Firsorder RL and RC Circuis 7. The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF sshifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationEcon107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)
I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following Kvariable regression
More informationTheory of! Partial Differential Equations!
hp://www.nd.edu/~gryggva/cfdcourse/! Ouline! Theory o! Parial Dierenial Equaions! Gréar Tryggvason! Spring 011! Basic Properies o PDE!! Quasilinear Firs Order Equaions!  Characerisics!  Linear and
More informationTHE BERNOULLI NUMBERS. t k. = lim. = lim = 1, d t B 1 = lim. 1+e t te t = lim t 0 (e t 1) 2. = lim = 1 2.
THE BERNOULLI NUMBERS The Bernoulli numbers are defined here by he exponenial generaing funcion ( e The firs one is easy o compue: (2 and (3 B 0 lim 0 e lim, 0 e ( d B lim 0 d e +e e lim 0 (e 2 lim 0 2(e
More information1 Review of ZeroSum Games
COS 5: heoreical Machine Learning Lecurer: Rob Schapire Lecure #23 Scribe: Eugene Brevdo April 30, 2008 Review of ZeroSum Games Las ime we inroduced a mahemaical model for wo player zerosum games. Any
More informationTHE 2BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI
THE 2BODY PROBLEM ROBERT J. VANDERBEI ABSTRACT. In his shor noe, we show ha a pair of ellipses wih a common focus is a soluion o he 2body problem. INTRODUCTION. Solving he 2body problem from scrach
More informationContinuous Time. TimeDomain System Analysis. Impulse Response. Impulse Response. Impulse Response. Impulse Response. ( t) + b 0.
TimeDomain Sysem Analysis Coninuous Time. J. Robers  All Righs Reserved. Edied by Dr. Rober Akl 1. J. Robers  All Righs Reserved. Edied by Dr. Rober Akl 2 Le a sysem be described by a 2 y ( ) + a 1
More informationSection 4.4 Logarithmic Properties
Secion. Logarihmic Properies 59 Secion. Logarihmic Properies In he previous secion, we derived wo imporan properies of arihms, which allowed us o solve some asic eponenial and arihmic equaions. Properies
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationLaplace transfom: ttranslation rule , Haynes Miller and Jeremy Orloff
Laplace ransfom: ranslaion rule 8.03, Haynes Miller and Jeremy Orloff Inroducory example Consider he sysem ẋ + 3x = f(, where f is he inpu and x he response. We know is uni impulse response is 0 for
More informationMath 4600: Homework 11 Solutions
Mah 46: Homework Soluions Gregory Handy [.] One of he wellknown phenomenological (capuring he phenomena, bu no necessarily he mechanisms) models of cancer is represened by Gomperz equaion dn d = bn ln(n/k)
More informationIMPLICIT AND INVERSE FUNCTION THEOREMS PAUL SCHRIMPF 1 OCTOBER 25, 2013
IMPLICI AND INVERSE FUNCION HEOREMS PAUL SCHRIMPF 1 OCOBER 25, 213 UNIVERSIY OF BRIISH COLUMBIA ECONOMICS 526 We have exensively sudied how o solve sysems of linear equaions. We know how o check wheher
More informationwhere the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).
Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness
More informationSection 5: Chain Rule
Chaper The Derivaive Applie Calculus 11 Secion 5: Chain Rule There is one more ype of complicae funcion ha we will wan o know how o iffereniae: composiion. The Chain Rule will le us fin he erivaive of
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationAPPM 2360 Homework Solutions, Due June 10
2.2.2: Find general soluions for he equaion APPM 2360 Homework Soluions, Due June 10 Soluion: Finding he inegraing facor, dy + 2y = 3e µ) = e 2) = e 2 Muliplying he differenial equaion by he inegraing
More informationLecture 13 RC/RL Circuits, Time Dependent Op Amp Circuits
Lecure 13 RC/RL Circuis, Time Dependen Op Amp Circuis RL Circuis The seps involved in solving simple circuis conaining dc sources, resisances, and one energysorage elemen (inducance or capaciance) are:
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se #2 Wha are ConinuousTime Signals??? Reading Assignmen: Secion. of Kamen and Heck /22 Course Flow Diagram The arrows here show concepual flow beween ideas.
More informationSection 4.4 Logarithmic Properties
Secion. Logarihmic Properies 5 Secion. Logarihmic Properies In he previous secion, we derived wo imporan properies of arihms, which allowed us o solve some asic eponenial and arihmic equaions. Properies
More informationLet ( α, β be the eigenvector associated with the eigenvalue λ i
ENGI 940 4.05  Sabiliy Analysis (Linear) Page 4.5 Le ( α, be he eigenvecor associaed wih he eigenvalue λ i of he coefficien i i) marix A Le c, c be arbirary consans. a b c d Case of real, disinc, negaive
More information1 st order ODE Initial Condition
Mah33 Chapers 11 s Order ODE Sepember 1, 17 1 1 s order ODE Iniial Condiion f, sandard form LINEAR NONLINEAR,, p g differenial form M x dx N x d differenial form is equivalen o a pair of differenial
More information2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS
Andrei Tokmakoff, MIT Deparmen of Chemisry, 2/22/2007 217 2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mahemaical formulaion of he dynamics of a quanum sysem is no unique. So far we have described
More informationIn this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should
Cambridge Universiy Press 97836600337 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationMatrix Versions of Some Refinements of the ArithmeticGeometric Mean Inequality
Marix Versions of Some Refinemens of he ArihmeicGeomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More informationTHE MATRIXTREE THEOREM
THE MATRIXTREE THEOREM 1 The MarixTree Theorem. The MarixTree Theorem is a formula for he number of spanning rees of a graph in erms of he deerminan of a cerain marix. We begin wih he necessary graphheoreical
More information5.1  Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1  Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationMath 527 Lecture 6: HamiltonJacobi Equation: Explicit Formulas
Mah 527 Lecure 6: HamilonJacobi Equaion: Explici Formulas Sep. 23, 2 Mehod of characerisics. We r o appl he mehod of characerisics o he HamilonJacobi equaion: u +Hx, Du = in R n, u = g on R n =. 2 To
More informationEECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and FirstOrder Linear Circuits
EEE25 ircui Analysis I Se 4: apaciors, Inducors, and FirsOrder inear ircuis Shahriar Mirabbasi Deparmen of Elecrical and ompuer Engineering Universiy of Briish olumbia shahriar@ece.ubc.ca Overview Passive
More informationAnd the solution to the PDE problem must be of the form Π 1
5. SelfSimilar Soluions b Dimensional Analsis Consider he diffusion problem from las secion, wih poinwise release (Ref: Bluman & Cole, 2.3): c = D 2 c x + Q 0δ(x)δ() 2 c(x,0) = 0, c(±,) = 0 Iniial release
More informationt, x 2 t above, using inputoutput modeling. Assume solute concentration is uniform in each tank. If x 1 0 = b 1
Mah 2254 Fri Apr 4 7 Sysems of differenial equaions  o model mulicomponen sysems via comparmenal analysis: hp://enwikipediaorg/wiki/mulicomparmen_model Here's a relaively simple 2ank problem o illusrae
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More informationTHE BELLMAN PRINCIPLE OF OPTIMALITY
THE BELLMAN PRINCIPLE OF OPTIMALITY IOANID ROSU As I undersand, here are wo approaches o dynamic opimizaion: he Ponrjagin Hamilonian) approach, and he Bellman approach. I saw several clear discussions
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a nonconsan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x componen of
More informationCERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS
SARAJEVO JOURNAL OF MATHEMATICS Vol.10 (22 (2014, 67 76 DOI: 10.5644/SJM.10.1.09 CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ Absrac. This paper presens sufficien
More informationApplication 5.4 Defective Eigenvalues and Generalized Eigenvectors
Applicaion 5.4 Defecive Eigenvalues and Generalized Eigenvecors The goal of his applicaion is he soluion of he linear sysems like where he coefficien marix is he exoic 5by5 marix x = Ax, (1) 9 11 21
More informationChapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws
Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species
More informationComparison between the Discrete and Continuous Time Models
Comparison beween e Discree and Coninuous Time Models D. Sulsky June 21, 2012 1 Discree o Coninuous Recall e discree ime model Î = AIS Ŝ = S Î. Tese equaions ell us ow e populaion canges from one day o
More informationEE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
EE 35 Noes Gürdal Arslan CLASS (Secions..2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of
More informationTraveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
More informationSTATESPACE MODELLING. A mass balance across the tank gives:
B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STESPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing
More information13.3 Term structure models
13.3 Term srucure models 13.3.1 Expecaions hypohesis model  Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)
More informationNavneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi
Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x componen of he velociy is a linear funcion (Figure 4.8(a)),
More informationInstructor: Barry McQuarrie Page 1 of 5
Procedure for Solving radical equaions 1. Algebraically isolae one radical by iself on one side of equal sign. 2. Raise each side of he equaion o an appropriae power o remove he radical. 3. Simplify. 4.
More informationHomework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
More information9231 FURTHER MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Level MARK SCHEME for he May/June series 9 FURTHER MATHEMATICS 9/ Paper, maximum raw mark This mark scheme is published as an aid o eachers and candidaes,
More informationGENERALIZATION OF THE FORMULA OF FAA DI BRUNO FOR A COMPOSITE FUNCTION WITH A VECTOR ARGUMENT
Inerna J Mah & Mah Sci Vol 4, No 7 000) 48 49 S0670000970 Hindawi Publishing Corp GENERALIZATION OF THE FORMULA OF FAA DI BRUNO FOR A COMPOSITE FUNCTION WITH A VECTOR ARGUMENT RUMEN L MISHKOV Received
More information4.1  Logarithms and Their Properties
Chaper 4 Logarihmic Funcions 4.1  Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,
More informationME 452 Fourier Series and Fourier Transform
ME 452 Fourier Series and Fourier ransform Fourier series From Joseph Fourier in 87 as a resul of his sudy on he flow of hea. If f() is almos any periodic funcion i can be wrien as an infinie sum of sines
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationA Note on Superlinear AmbrosettiProdi Type Problem in a Ball
A Noe on Superlinear AmbroseiProdi Type Problem in a Ball by P. N. Srikanh 1, Sanjiban Sanra 2 Absrac Using a careful analysis of he Morse Indices of he soluions obained by using he Mounain Pass Theorem
More informationLinear Surface Gravity Waves 3., Dispersion, Group Velocity, and Energy Propagation
Chaper 4 Linear Surface Graviy Waves 3., Dispersion, Group Velociy, and Energy Propagaion 4. Descripion In many aspecs of wave evoluion, he concep of group velociy plays a cenral role. Mos people now i
More informationPositive continuous solution of a quadratic integral equation of fractional orders
Mah. Sci. Le., No., 97 (3) 9 Mahemaical Sciences Leers An Inernaional Journal @ 3 NSP Naural Sciences Publishing Cor. Posiive coninuous soluion of a quadraic inegral equaion of fracional orders A. M.
More information2.4 Cuk converter example
2.4 Cuk converer example C 1 Cuk converer, wih ideal swich i 1 i v 1 2 1 2 C 2 v 2 Cuk converer: pracical realizaion using MOSFET and diode C 1 i 1 i v 1 2 Q 1 D 1 C 2 v 2 28 Analysis sraegy This converer
More informationEigenvalues and Eigenvectors. Eigenvalues and Eigenvectors. Initialization
Eigenvalues and Eigenvecors Iniializaion ClearAll@"Global` "D; Off@General::spell, General::spellD; Eigenvalues and Eigenvecors We will now review some ideas from linear algebra. Proofs of he heorems are
More informationExpert Advice for Amateurs
Exper Advice for Amaeurs Ernes K. Lai Online Appendix  Exisence of Equilibria The analysis in his secion is performed under more general payoff funcions. Wihou aking an explici form, he payoffs of he
More informationf(s)dw Solution 1. Approximate f by piecewise constant leftcontinuous nonrandom functions f n such that (f(s) f n (s)) 2 ds 0.
Advanced Financial Models Example shee 3  Michaelmas 217 Michael Tehranchi Problem 1. Le f : [, R be a coninuous (nonrandom funcion and W a Brownian moion, and le σ 2 = f(s 2 ds and assume σ 2
More informationLaplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)
Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer PierreSimon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful
More informationWeek #13  Integration by Parts & Numerical Integration Section 7.2
Week #3  Inegraion by Pars & Numerical Inegraion Secion 7. From Calculus, Single Variable by HughesHalle, Gleason, McCallum e. al. Copyrigh 5 by John Wiley & Sons, Inc. This maerial is used by permission
More information1. An introduction to dynamic optimization  Optimal Control and Dynamic Programming AGEC
This documen was generaed a :45 PM 8/8/04 Copyrigh 04 Richard T. Woodward. An inroducion o dynamic opimizaion  Opimal Conrol and Dynamic Programming AGEC 63704 I. Overview of opimizaion Opimizaion is
More information1. An introduction to dynamic optimization  Optimal Control and Dynamic Programming AGEC
This documen was generaed a :37 PM, 1/11/018 Copyrigh 018 Richard T. Woodward 1. An inroducion o dynamic opimiaion  Opimal Conrol and Dynamic Programming AGEC 64018 I. Overview of opimiaion Opimiaion
More informationLecture #31, 32: The OrnsteinUhlenbeck Process as a Model of Volatility
Saisics 441 (Fall 214) November 19, 21, 214 Prof Michael Kozdron Lecure #31, 32: The OrnseinUhlenbeck Process as a Model of Volailiy The OrnseinUhlenbeck process is a di usion process ha was inroduced
More informationLecture 21 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure  Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationBernoulli numbers. Francesco Chiatti, Matteo Pintonello. December 5, 2016
UNIVERSITÁ DEGLI STUDI DI PADOVA, DIPARTIMENTO DI MATEMATICA TULLIO LEVICIVITA Bernoulli numbers Francesco Chiai, Maeo Pinonello December 5, 206 During las lessons we have proved he Las Ferma Theorem
More informationWe just finished the ErdősStone Theorem, and ex(n, F ) (1 1/(χ(F ) 1)) ( n
Lecure 3  KövariSósTurán Theorem Jacques Versraëe jacques@ucsd.edu We jus finished he ErdősSone Theorem, and ex(n, F ) ( /(χ(f ) )) ( n 2). So we have asympoics when χ(f ) 3 bu no when χ(f ) = 2 i.e.
More informationBoyce/DiPrima/Meade 11 th ed, Ch 3.1: 2 nd Order Linear Homogeneous EquationsConstant Coefficients
Boce/DiPrima/Meade h ed, Ch 3.: nd Order Linear Homogeneous EquaionsConsan Coefficiens Elemenar Differenial Equaions and Boundar Value Problems, h ediion, b William E. Boce, Richard C. DiPrima, and Doug
More informationVanishing Viscosity Method. There are another instructive and perhaps more natural discontinuous solutions of the conservation law
Vanishing Viscosiy Mehod. There are anoher insrucive and perhaps more naural disconinuous soluions of he conservaion law (1 u +(q(u x 0, he so called vanishing viscosiy mehod. This mehod consiss in viewing
More informationQuestion 1: Question 2: Topology Exercise Sheet 3
Topology Exercise Shee 3 Prof. Dr. Alessandro Siso Due o 14 March Quesions 1 and 6 are more concepual and should have prioriy. Quesions 4 and 5 admi a relaively shor soluion. Quesion 7 is harder, and you
More informationCHAPTER 6: FIRSTORDER CIRCUITS
EEE5: CI CUI T THEOY CHAPTE 6: FISTODE CICUITS 6. Inroducion This chaper considers L and C circuis. Applying he Kirshoff s law o C and L circuis produces differenial equaions. The differenial equaions
More informatione 2t u(t) e 2t u(t) =?
EE : Signals, Sysems, and Transforms Fall 7. Skech he convoluion of he following wo signals. Tes No noes, closed book. f() Show your work. Simplify your answers. g(). Using he convoluion inegral, find
More informationln y t 2 t c where c is an arbitrary real constant
SOLUTION TO THE PROBLEM.A y y subjec o condiion y 0 8 We recognize is as a linear firs order differenial equaion wi consan coefficiens. Firs we sall find e general soluion, and en we sall find one a saisfies
More information7 Wave Equation in Higher Dimensions
7 Wave Equaion in Highe Dimensions We now conside he iniialvalue poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,
More informationChapter 2 The Derivative Applied Calculus 107. We ll need a rule for finding the derivative of a product so we don t have to multiply everything out.
Chaper The Derivaive Applie Calculus 107 Secion 4: Prouc an Quoien Rules The basic rules will le us ackle simple funcions. Bu wha happens if we nee he erivaive of a combinaion of hese funcions? Eample
More informationLecture 4: Processes with independent increments
Lecure 4: Processes wih independen incremens 1. A Wienner process 1.1 Definiion of a Wienner process 1.2 Reflecion principle 1.3 Exponenial Brownian moion 1.4 Exchange of measure (Girsanov heorem) 1.5
More informationarxiv:math/ v1 [math.nt] 3 Nov 2005
arxiv:mah/0511092v1 [mah.nt] 3 Nov 2005 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETAFUNCTION D. A. GOLDSTON AND S. M. GONEK Absrac. Le πs denoe he argumen of he Riemann zeafuncion a he poin 1 + i. Assuming
More informationLecture #8 Redfield theory of NMR relaxation
Lecure #8 Redfield heory of NMR relaxaion Topics The ineracion frame of reference Perurbaion heory The Maser Equaion Handous and Reading assignmens van de Ven, Chapers 6.2. Kowalewski, Chaper 4. Abragam
More informationASTR415: Problem Set #5
ASTR45: Problem Se #5 Curran D. Muhlberger Universi of Marland (Daed: April 25, 27) Three ssems of coupled differenial equaions were sudied using inegraors based on Euler s mehod, a fourhorder RungeKua
More informationHomework 4 (Stats 620, Winter 2017) Due Tuesday Feb 14, in class Questions are derived from problems in Stochastic Processes by S. Ross.
Homework 4 (Sas 62, Winer 217) Due Tuesday Feb 14, in class Quesions are derived from problems in Sochasic Processes by S. Ross. 1. Le A() and Y () denoe respecively he age and excess a. Find: (a) P{Y
More informationFourier Series & The Fourier Transform. Joseph Fourier, our hero. Lord Kelvin on Fourier s theorem. What do we want from the Fourier Transform?
ourier Series & The ourier Transfor Wha is he ourier Transfor? Wha do we wan fro he ourier Transfor? We desire a easure of he frequencies presen in a wave. This will lead o a definiion of he er, he specru.
More information