Math 334 Midterm III KEY Fall 2006 sections 001 and 004 Instructor: Scott Glasgow

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1 Math 334 Midterm III KEY Fall 6 sections 1 and 4 Instructor: Scott Glasgow Please do NO write on this exam No credit will be given for such work Rather write in a blue book, or on your own paper, preferably engineering Warning: check your solutions to each problem via a method independent of the one used to obtain your initial solution

2 1 1 Solve the following initial value problem in terms of the convolution integral: y + 9 y = g( t); y() = y () = (11) 15 points Laplace transformation of (11) gives [ ] + 9 [ ] = [ ] + 9 [ ] = [ ] sl y L y L y L y Lg 1 1 L[ y] = L [ g] = L sin ( 3 ) g s t 1 y() t = sin( 3 ) g () t = sin( 3( t τ )) g( τ) dτ 3 3 (1) Find the general solution of the following system: 1 1 x = 4 1 x (13) points he general solution of (13) is λ1t λt x = x() t = cξ e + c ξ e, (14) 1 1 where the ξ s and λ s are independent eigenvectors and distinct eigenvalues of the matrix in (13):

3 λ 1 = = 4 1 λ ξ ξ unless λ 1 = det 4 1 λ = λ = 1± = 3, 1 = : λ, λ 1 ( ) λ (15) So = , ξ = = = 4 ξ ξ ξ ( ) = ξ = 4 1 ( 1) = = 4 ξ ξ ξ (16) hus, explicitly, (14) is x 1 3t 1 t = x () t = c1 e + c e (17) 3 Show that L[ f]( s) = st e f() t dt (18) s 1 e for s>, provided f ( t ) f( t) 1 points + = for all [, ] st t + (and provided e f() t dtexists) We have

4 3 + ( n+ 1) st st s( t+ n ) L[ f ]( s): = e f ( t) dt = e f ( t) dt = e f ( t + n ) dt n n= n= sn st st s st 1 = e e f() t dt = e f() t dt e = e f() t dt 1 e = n ( ) ( ) n= n= st e f() t dt s 1 e s (19) 4 Find a real-valued representation of the general solution of the following system: 1 5 x = 3 x (11) 5 points he general solution of (11) can be expressed as λ1t λt x = x() t = cξ e + c ξ e, (111) 1 1 where the ξ s and λ s are independent eigenvectors and distinct eigenvalues of the matrix in (11): λ 5 = = 1 3 λ ξ ξ unless So λ 5 = det = ( λ 1)( λ + 3) + 5= λ + λ + = ( λ + 1) i 3 λ λ = 1± i = 1 + i, i = : λ, λ 1 (11) ( ) 1+ i 5 i 5 i 5 i 5 + i = ξ1 = , 1 3 ( 1 i) 1 i ξ = = = i 5 ξ + + ξ ξ ( ) i 5 + i 5 + i 5 + i 5 i = ξ = ( 1 i) 1 i ξ = = = = i 5 ξ + ξ ξ ξ (113)

5 4 hus, explicitly, (111) is x + i ( + 1 it ) i ( 1 it ) = x () t = c1 e + c e (114) As per the usual theory, we can find a real-valued representation by finding the real and imaginary parts of either of the above complex-valued solutions: + i ( + 1 it ) + i t cost sin t t cost+ sin t t x1() t = e = e ( cost+ isin t) = e + i e, cost sint (115) whence a real-valued representation of the general solution is x cost sint t cost+ sint t () t = c1 e + c e cost sin t (116) 5 Find the fundamental matrix of solutions Φ =Φ( t) to the above problem that has 1 the property that Φ () = I = 1 1 points A fundamental matrix of solutions Ψ =Ψ ( t), one not necessarily having the desired property, can be found from the above general solution (116): t cost sin t cost+ sin t Ψ () t = e cost sin t (117) he desired fundamental matrix Φ =Φ( t) can be obtained from Ψ =Ψ( t) via t cost sint cost+ sint 1 Φ=Φ ( t) =Ψ( t) Ψ () = e cost sin t 1 (118) t cost sin t cost+ sin t 1 t cost+ sin t 5sin t = e = e cost sin t 1 sin t cost sin t

6 5 6 Solve the initial value problem given by the system of problem 4 and the initial data 7 points x () = (119) (Hint: rather than reinventing the wheel, just use the fundamental matrix of problem 5) Using the fundamental matrix of problem 5 we have x t cost+ sin t 5sin t t cost sin t () t =Φ () t x () = e = e sin t cost sin t cost (1) 7 Calculate (Hint: use the result of problem 5) 1 5 π 3 e (11) 5 points We have, from problem 5, e 1 5 π π π π π π e 1 3 cos + sin 5sin =Φ ( π ) = e π sinπ cosπ sinπ = e (1) 8 Find a representation of the general solution of the system 3 9 x = 3 x (13) points

7 6 he matrix in (13) has a repeated eigenvalue with only one eigenvector Hence the general solution is of the form λt 1 ( ) λt x = x() t = cξe + c ξt+ η e (14) where ξ is an eigenvector and ηis an associated pseudo eigenvector: 3 λ λ ξ = ξ = unless 3 λ 9 = det = λ + 3λ 3λ 9+ 9= λ 3 λ λ =,, (15) So =, 1 3 ξ = ξ ξ = 1 and η 1 3 η= ξ = = = 1 η η η (16) hus, explicitly, (14) is 3 3 3η x = x () t = c1 + c t+ η (17) 9 Find the fundamental matrix of solutions Φ =Φ( t) for the system of problem 8 1 that satisfies Φ () = I = 1 15 points

8 7 From (17) we have a fundamental matrix of solutions whence the one desired is 3 3t + 1 3η, (18) Ψ () t = t + η 3 3t 1 3η η + Φ () t = Ψ() t Ψ () = t + η η 3 3t + 1 3η η 1+ 3η 1+ 3t 9t = = t + η 1 3 t 1 3t (19) 1 Solve the initial value problem obtained from combining the differential equation of problem 8 with the initial data x () = 4 (13) (Hint: do not reinvent the wheel, but rather use the result from problem 9) 6 points From problem 9 we have x 1+ 3t 9t + 4t () t =Φ () t x () = = t 1 3t 4 44t (131)

KEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow

KEY. 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