Math 307 Spring, 2005 Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://www.math.iastate.edu/hentzel/class.307.05 ============================================================ Assignment: Solve the differential Equation / | x | | 1 -3 2 2 || x | | y | = | 0 -1 2 1 || y | | z | |-1 -3 4 2 || z | | w | | 0 -3 2 3 || w | April 8 5.7 Applications to differential Equations ----------------------------------------------------------- -1 Previous Assignment. Find a matrix P such that P A P is in Jordan Canonical Form. | 3 3 2 2 | A = | 0 2 1 0 | Det[A-xI] = (x-2)^4 | 1 1 3 2 | |-1 -2 -2 0 | x y z w RHS | 1 3 2 2 r | A-2I = | 0 0 1 0 s | | 1 1 1 2 t | |-1 -2 -2 -2 u | | 1 3 2 2 r | == | 0 0 1 0 s | | 0 -2 -1 0 -r+t | | 0 1 0 0 r+u | | 1 3 2 2 r | | 0 1 0 0 r+u | == | 0 0 1 0 s | | 0 -2 -1 0 -r+t | | 1 0 2 2 -2r-3u | | 0 1 0 0 r+u | == | 0 0 1 0 s | | 0 0 -1 0 r+2u+t | x y z w=a RHS | 1 0 0 2 -2r-2s-3u | | 0 1 0 0 r+u | == | 0 0 1 0 s | | 0 0 0 0 r+s+2u+t | r+s+2u+t = 0 so let t = -r-s-2u | r | So B = | s | |-r-s-2u | | u | | x | | -2r-2s-3u | | -2 | | y | = | r+u | + a | 0 | | z | | s | | 0 | | w | | 0 | | 1 | r=-2 s=0 u=1 r= 1 s=-1 u=0 r=0 s= 1 u=0 | -2 | | 1 | | 0 | |-2 | 0 <--- | 0 | <--- | -1 | <---------- | 1 | <------ | 0 | | 0 | | 0 | |-1 | | 1 | | 1 | | 0 | | 0 | | 0 | | -2 1 0 -2 | -1 | 2 1 0 0 | P = | 0 -1 1 0 | P A P = | 0 2 1 0 | | 0 0 -1 1 | | 0 0 2 1 | | 1 0 0 0 | | 0 0 0 2 | ============================================================== Find the Jordan Canonical form for the matrix | 4 4 4 4 | A = | -1 1 -1 -2 | | 1 1 3 2 | | -1 -2 -2 0 | Hint: the characteristic polynomial is (x-2)^4. x y z w [ A-2I | RHS] | 2 4 4 4 r | | -1 -1 -1 -2 s | | 1 1 1 2 t | | -1 -2 -2 -2 u | | 1 0 0 2 -r/2-2s | | 0 1 1 0 r/2+s | | 0 0 0 0 s + t | | 0 0 0 0 u+r/2 | Thus s = -t and r = -2u x y z=a w=b RHS | 1 0 0 2 u+2t | | 0 1 1 0 -u-t | | 0 0 0 0 0 | | 0 0 0 0 0 | This is the solution to | x | |-2u| (A-2I) | y | = |-t | | z | | t | | w | | u | | x | | u+2t | | 0 | | -2 | | y | = | -u-t | + a |-1 | + b | 0 | | z | | 0 | | 1 | | 0 | | w | | 0 | | 0 | | 1 | u=0 t=1 u=1 t=0 | 0 | | 2 | | -2 | | 1 | 0 <--- |-1 | <---- |-1 | 0 <----- | 0 | <----- | -1 | | 1 | | 0 | | 0 | | 0 | | 0 | | 0 | | 1 | | 0 | | 0 2 -2 1 | | 4 4 4 4 | -1 | 2 1 0 0 | P = |-1 -1 0 -1 | A = | -1 1 -1 -2 | P A P = | 0 2 0 0 |. | 1 0 0 0 | | 1 1 3 2 | | 0 0 2 1 | | 0 0 1 0 | | -1 -2 -2 0 | | 0 0 0 2 | --------------------------------------------------------- Solve the differential equation: / | x1| | 4 4 4 4 | | x1| | x2| = | -1 1 -1 -2 | | x2| | x3| | 1 1 3 2 | | x3| | x4| | -1 -2 -2 0 | | x4| | x1| | 0 2 -2 1 | | y1 | | x2| = |-1 -1 0 -1 | | y2 | | x3| | 1 0 0 0 | | y3 | | x4| | 0 0 1 0 | | y4 | The related equation is | y1 | | 2 1 0 0 | | y1 | | y2 | = | 0 2 0 0 | | y2 | | y3 | | 0 0 2 1 | | y3 | | y4 | | 0 0 0 2 | | y4 | Solution Solution Solution Solution | y1 | 2t | 1 | 2t |1+t| 2t | 0 | 2t | 0 | | y2 | = C1 e | 0 | + C2 e | 1 | + C3 e | 0 | + C4 e | 0 | | y3 | | 0 | | 0 | | 1 | |1+t| | y4 | | 0 | | 0 | | 0 | | 1 | The original equation has solution | x1 | | 0 2 -2 1 | | 1 1+t 0 0 | | C1 | 2t | x2 | = |-1 -1 0 -1 | | 0 1 0 0 | | C2 | e | x3 | | 1 0 0 0 | | 0 0 1 1+t | | C3 | | x4 | | 0 0 1 0 | | 0 0 0 1 | | C4 | / | x1 | | 0 2 -2 1 | | 2 3+2t 0 0 | | C1 | 2t | x2 | = |-1 -1 0 -1 | | 0 2 0 0 | | C2 | e | x3 | | 1 0 0 0 | | 0 0 2 3+2t| | C3 | | x4 | | 0 0 1 0 | | 0 0 0 2 | | C4 | / | x1 | | 0 4 -4 -4-4t | | C1 | 2t | x2 | = | -2 -5-2t 0 -2 | | C2 | e | x3 | | 2 3+2t 0 0 | | C3 | | x4 | | 0 0 2 3+2t | | C4 | | x1| | 4 4 4 4 || 0 2 -2 1 | 1 1+t 0 0 | | C1 | 2t A | x2| = | -1 1 -1 -2 ||-1 -1 0 -1 | 0 1 0 0 | | C2 | e | x3| | 1 1 3 2 || 1 0 0 0 | 0 0 1 1+t | | C3 | | x4| | -1 -2 -2 0 || 0 0 1 0 | 0 0 0 1 | | C4 | | x1| | 0 4 -4 0| | 1 1+t 0 0 | |C1| 2t A | x2| = |-2 -3 0 -2| | 0 1 0 0 | |C2| e | x3| | 2 1 0 0| | 0 0 1 1+t | |C3| | x4| | 0 0 2 1| | 0 0 0 1 | |C4| | x1| | 0 4 -4 -4-4t| |C1| 2t A | x2| = |-2 -5-2t 0 -2 | |C2| e | x3| | 2 3+2t 0 0 | |C3| | x4| | 0 0 2 3+2t| |C4| --------------------------------------------------------------------------