Math 102. Rumbos Spring Solutions to Assignment #8. Solution: The matrix, A, corresponding to the system in (1) is

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1 Math 12. Rumbos Spring Solutions to Assignmnt #8 1. Construct a fundamntal matrix for th systm { ẋ 2y ẏ x + y. (1 Solution: Th matrix, A, corrsponding to th systm in (1 is 2 A. (2 1 1 Th charactristic polynomial of th matrix A in (2 is which factors into p A (λ λ 2 λ 2, p A (λ (λ + 1(λ 2 so that, λ 1 1 and λ 2 2 ar th ignvalus of A. Nxt, w comput th corrsponding ignspacs. For λ 1 1, w solv th homognous systm Th augmntd matrix form of th systm in (3 is 1 2, 1 2 which rducs to 1 2 (A λ 1 Iv. (3 so that, th systm in (3, with λ 1 1, is quivalnt to th quation x 1 + 2x 2, which can b solvd to yild th span of th vctor 2 v 1 1 (4 as its solution spac.

2 Math 12. Rumbos Spring Nxt, w find an ignvctor, v 2, corrsponding to λ 2 2 by solving th homognous systm (A λ 2 Iv, (5 by rducing th augmntd matrix ( of th systm in (5 to 1 1 so that, th systm in (5, with λ 2 2, is quivalnt to th quation which can b solvd to yild x 1 x 2, v as an ignvctor corrsponding to λ 2 2. St Q [ v 1 that, W st whr v 2 ], whr v1 and v 2 ar givn in (4 and (6, rspctivly so Q ( (7 1 1 J Q 1 AQ, (8 Q , (9 1 2 It follows from (2, (8, (7 and (9 that 1 J. (1 2 Th fundamntal matrix corrsponding to th matrix J in (1 is thn t (t 2t, for all t R. (11 Th fundamntal matrix corrsponding to A is givn by (t Q (tq 1, for all t R,

3 Math 12. Rumbos Spring whr Q, (t and Q 1 ar givn in (7, (11 and (9, rspctivly. W thn obtain that 2 1 t (t 1 1 2t so that, (t ( ( t t 2t 2 2t ( 2 t + 2t 2 t + 2 2t t + 2t t + 2 2t, for all t R. 2. Construct a fundamntal matrix for th systm { ẋ x + 3y ẏ 2x + 6y. (12 Solution: Th matrix, A, corrsponding to th systm in (12 is 1 3 A. ( Th charactristic polynomial of th matrix A in (13 is which factors into p A (λ λ 2 7λ, p A (λ λ(λ 7 so that, λ 1 and λ 2 7 ar th ignvalus of A. Nxt, w comput th corrsponding ignspacs. For λ 1, w solv th homognous systm Th augmntd matrix form of th systm in (14 is 1 3, 2 6 (A λ 1 Iv. (14

4 Math 12. Rumbos Spring which rducs to 1 3 so that, th systm in (14, with λ 1, is quivalnt to th quation x 1 + 3x 2, which can b solvd to yild th span of th vctor 3 v 1 1 (15 as its solution spac. Nxt, w find an ignvctor, v 2, corrsponding to λ 2 7 by solving th homognous systm (A λ 2 Iv, (16 by rducing th augmntd matrix ( of th systm in (16 to 2 1 so that, th systm in (16, with λ 2 7, is quivalnt to th quation which can b solvd to yild 2x 1 x 2, v as an ignvctor corrsponding to λ 2 7. St Q [ v 1 that, W st (17 v 2 ], whr v1 and v 2 ar givn in (16 and (17, rspctivly so Q 3 1. ( J Q 1 AQ, (19

5 Math 12. Rumbos Spring whr Q , (2 1 3 It follows from (13, (18, (19 and (2 that J. (21 7 Th fundamntal matrix corrsponding to th matrix J in (21 is thn 1 (t 7t, for all t R. (22 Th fundamntal matrix corrsponding to A is givn by (t Q (tq 1, for all t R, whr Q, (t and Q 1 ar givn in (18, (22 and (2, rspctivly. W thn obtain that (t 1 2 7t so that, (t t 3 7t ( 6 + 7t t t t, for all t R. 3. Construct a fundamntal matrix for th systm { ẋ 2x + y ẏ x + 4y. (23 Solution: Th matrix, A, corrsponding to th systm in (23 is 2 1 A. (24 1 4

6 Math 12. Rumbos Spring Th charactristic polynomial of th matrix A in (24 is which factors into p A (λ λ 2 6λ + 9, p A (λ (λ 3 2 so that, λ 3 is th only ignvalu of A. Nxt, w comput th ignspac corrsponding to λ 3, by solving th homognous systm (A λiv, (25 with λ 3. Th augmntd matrix form of th systm in (25 is 1 1, 1 1 which rducs to 1 1 so that, th systm in (25, with λ 3, is quivalnt to th quation x 1 x 2, which can b solvd to yild th span of th vctor 1 v 1 1 (26 as its solution spac. Hnc, thr is no basis for R 2 mad up of ignvctors of A thrfor, A is not diagonalizabl. Nxt, w find a solution, v 2, of th nonhomognous systm by rducing th augmntd matrix ( of th systm in (27 to (A λiv v 1, (27

7 Math 12. Rumbos Spring so that, th systm in (27, with λ 3, is quivalnt to th quation which can b solvd to yild x1 t x 2 Taking t in (28 yilds St Q [ v 1 that, Nxt, st whr x 1 x 2 1, v 2 1, for t R. (28 1. (29 v 2 ], whr v1 and v 2 ar givn in (26 and (29, rspctivly so Q 1 1. (3 1 J Q 1 AQ, (31 Q 1 1. ( It follows from (24, (31, (3 and (32 that 3 1 J. (33 3 Th fundamntal matrix, (t, corrsponding to th matrix J in (33 is givn by 3t t (t 3t 3t, for all t R. (34 Th fundamntal matrix corrsponding to th matrix A in (24 is thn givn by (t Q (tq 1, for all t R, whr Q, (t and Q 1 ar givn in (3, (34 and (32, rspctivly. W thn obtain that 1 1 3t t (t 3t 1 1 3t 1 1 ( ( t 3t 3t + t 3t 3t 3t

8 Math 12. Rumbos Spring so that, (t t t + 3t t 3t t 3t 3t + t 3t, for all t R. 4. Lt dnot th fundamntal matrix of th two dimnsional linar systm (ẋ x A, (35 ẏ y whr A is a 2 2 matrix with ral ntris. Show that is invrtibl and [ (t] 1 ( t, for all t R. (36 Solution: Any 2 2 matrix, A is similar to on of th matrics λ1 J, (37 λ 2 or or J λ 1, (38 λ α β J. (39 β α That is, thr xists and invrtibl 2 2 matrix, Q, such that Q 1 AQ J, (4 whr J is in on of th standard forms in (37, or(38, or (39. To ach of th matrics, J, in (37, or(38, or (39, thr corrsponds a fundamntal matrix, (t, givn by λ 1 t (t λ 2t, for t R, (41 or λt t (t λt λt, for t R, (42

9 Math 12. Rumbos Spring or rspctivly. (t αt cos βt αt sin βt αt sin βt αt, for t R, (43 cos βt Th fundamntal matrix corrsponding to th systm in (35 is thn givn by (t Q (tq 1, (44 whr J is on of th matrics in (37, or (38, or (39, that is similar to A by mans of th quation in (4. In ordr to stablish th rsult in (36, w will first show that it is tru for th fundamntal matrics, (t, in ach of th thr cass in (41, (42 and (43 that is, w will first show that [ (t] 1 ( t, for all t R, (45 for ach of th thr cass in (41 or (42 or (43. For th cas in which (t is givn by (41, w comput [ (t] 1 λ 1 t λ 2t ( λ 1 ( t ( t, for all t R and so (45 is vrifid in this cas. λ 2( t For th cas in which (t is givn by (42, w comput [ (t] 1 1 λt t λt 2λt λt λt t λt λt λ( t ( t λ( t ( t, λ( t

10 Math 12. Rumbos Spring for all t R and so (45 is vrifid in this cas. For th cas in which (t is givn by (42, w comput [ (t] 1 1 αt cos βt αt sin βt 2αt αt sin βt αt cos βt ( αt cos( βt αt sin( βt αt sin( βt αt cos( βt ( α( t cos(β( t α( t sin(β( t α( t sin(β( t α( t cos(β( t ( t, for all t R and so (45 is vrifid in this cas. Finally, us (44 and (45 to comput [ (t] 1 [Q (tq 1 ] 1 [Q 1 ] 1 [ (t] 1 Q 1 Q ( tq 1 ( t for all t R and so (36 is provd. 5. Lt A and b as in Problm 4. Show that (t + τ (t (τ, for all t, τ R. (46 Solution: W procd as in th solution of Problm 4 by vrifying first that (t + τ (t (τ, for all t, τ R, (47 for ach of th cass in (41, (42 and (43.

11 Math 12. Rumbos Spring For th cas in which (t is givn by (41, w comput λ 1 (t+τ (t + τ λ 2(t+τ ( λ 1 t λ 1τ λ2t λ 2τ λ 1 t λ 1 τ λ 2t λ 2τ (t (τ, for all t R and so (47 is vrifid in this cas. For th cas in which (t is givn by (42, w comput λ(t+τ (t + τ (t + τ λ(t+τ λ(t+τ λt λτ t λt λτ + τ λt λτ λt λτ λt t λt λτ τ λτ λt (t (τ, for all t R and so (47 is vrifid in this cas. λτ For th cas in which (t is givn by (43, w comput (t + τ α(t+τ cos β(t + τ α(t+τ sin β(t + τ α(t+τ sin β(t + τ α(t+τ cos β(t + τ cos β(t + τ sin β(t + τ αt ατ sin β(t + τ cos β(t + τ cos βt cos βτ sin βt sin βτ sin βt cos βτ cos βt sin βτ αt ατ, sin βt cos βτ + cos βt sin βτ cos βt cos βτ sin βt sin βτ whr w hav usd th trigonomtric idntitis cos(θ + ψ cos θ cos ψ sin θ sin ψ

12 Math 12. Rumbos Spring and sin(θ + ψ sin θ cos ψ + cos θ sin ψ. W thn hav that (t + τ cos βt sin βt cos βτ sin βτ αt ατ sin βt cos βt sin βτ cos βτ ( cos βt sin βt cos βτ αt sin βt cos βt ατ sin βτ (t (τ, for all t R and so (47 is vrifid in this cas. W now us th xprssion for in (44, and (47 to comput (t + τ Q (t + τq 1 Q (t (τq 1 Q (tq 1 Q (τq 1 (t (τ, sin βτ cos βτ for all t and τ in R, which stablishs (46.