18.034, Honors Differential Equations Prof. Jason Starr Lecture 26 4/9/04 1. Spent about ½ lecture working through the linear system of a pair of masses connected by a spring (of equilibrium displacement L) L)-X-k(x- ”xmbaaa= L)-X-k(x- ”xmbaba= Introduce , , X= , A='xvaa= 'XVbb=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛bbaavxvx⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡−−=001000000010bbaamkmkmkmk, F=⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡−=bamkLmkL00. Then . Physics (intuition suggests introducing FAx x' +=babammmm+=M, baabxmMxmMy +=1, baabxmMxmMyV +== '11, ba2x-xy=, ba11V-V 'y'V ==. Then G+= Byy' where ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡−=000100000000010 Bmk, ⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡=MkL000G. Now (y1,V1) is decoupled from (y2,V2). 2. We paused at this point and solved the system by usual 2nd order method. But then we continued to ‘diagonalize’ (y2,V2). Implicit C.O.V. : → 212212ziziVzz yωω+=+=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+−+⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=MLiMLizii2200z'ωωωω, where Mk=ω This is now completely diagonalized: ⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡−+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡−mLmLezeztvyzzvytiti2200100001000010000100010,20,10,10,12111ωω. 18.034, Honors Differential Equations Prof. Jason Starr Page 1 of 3Back-substituting: ⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡−+⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡−+⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡−−+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡−0010100010010101010,20,10,10,1batbbaatibbaabbaamLmLemMimMmMimMzemMimMmMimMzttvyvxvxωωωωωω Need to choose z1,0 and z2,0 to be complex conjugate to get a real solution. eigenval = 0 eigenvector= eigenval = 0 gen. eigenvector eigenval = iω eigenvect eigenval = -iω eigenvector Particular sol’n. 3. Discussed the general eigenvector decomp. method for solving : If (λ,V) is an eigenvalue/ eigenvector pair, then is a sol’n. Ay y' =tve y(t)λ= Defined eigenvalues + eigenvectors. Defined the char. poly, A)-Idet(λ. Saw that the eigenvalues are precisely the roots of A)-Idet(λ. We did not yet define/ discuss generalized eigenspaces (although I did mention the term in the solution of the 2-mass-spring problem). 18.034, Honors Differential Equations Prof. Jason Starr Page 2 of 31. One or two problems reducing higher order constant coeff. linear systems to 1st or order linear systems (students seemed fuzzy on this). Examples: (a) : te y 3y'- 3y"- 'y" =+ , +⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡−= y330100010y'⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡te00(b) 3524231211''yxyxyxyxyx===== x⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=0010110000010000010000010 x' 2. Maybe one problem going in the other direction: Example: : ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎦⎤⎢⎣⎡21'212112yyyy111211112'-4y2yy'y'2y-”y'-2yyy+===3tt23tt1111Be-AeyBeAey03y4y-”y+=+==+ y1”=y2y2”=y3y3’=y1+y2y1= y y2= y’ y3= y” 3. Several diagonalizing/ eigenvalue- eigenvector problems e.g. (a) , ; ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−31134)2)(( 862++=++λλλλ2- =λ, , ⎥⎦⎤⎢⎣⎡11 4- =λ, ⎥⎥⎦⎤⎢⎢⎣⎡−11 (b) . ⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡300320321By exercise (18) (This week’s Pset), 3)-2)(-1)(-(λλλ: 1 =λ, ; ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡0012 =λ, ; ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡0123 =λ, . ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡269 (c) , ; ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−211021)( +λ1- =λ, . ⎥⎦⎤⎢⎣⎡1118.034, Honors Differential Equations Prof. Jason Starr Page 3 of
View Full Document
Unlocking...