Question

4. Solve the nonhomogeneous linear system of differential equations

2. Solve the nonhomogeneous linear system of anerential equations () u-9" (). 3. Solve the homogeneous linear system of differential equations 1 ( 2 ) uten ( 46 ) + ( ). 4. Solve the nonhomogeneous linear system of differential equations 43,742 cos(46) - 4 sin(40) (10 5 cos(40) ) +847, 7 4cos(46) + 2 sin(40) 5 sin(46) 5. Solve the initial value problem for the nonhomogeneous linear system of differential equations 413,762 cos(41) - 4 sin(4) 5 cos(46) 847,744 cos(46) + 2 sin(46) 5 sin(40) u(0) – ( 13 ) 6. Solve the nonhomogeneous system of differential equations

Answer 1

The given system is 9 - 4 15 s + 2 cost - 4shat ut 4 3 7 t/ 5 Costt t ( 4 cos tt + zsinat 5 sin 4t t a t си den- we The system is the form i = Au + f(t) first we will find eigenvalues and eigen- vector of A LA RI) = 0 19-2 -4 -0. 22 - 14h+65=0 h = 7+ 4i 612 = 7+41, for eigen vector [A-2, Iv=0 *2]CJ 10

1 2-41 -2-4i Trai 5 -2-4i) R - 5R, 15 R 1 0 -2-4 i 12:17 0 JL x = 2+tin 5 and n = 12 Then we = (2+ 4i x 5 x2 싸혼 5 ro4d Then die similarly for h = 7ti (2-407 0 0

Now, we define alt) = cat o 8+4i) to (2+417 S = y con cos at isinat), 1:36] 2 sintt 4t 7th (204t 7 rasinate 5 cos tt 5 sin 4t tift cost t] } 2sin4 t + 4 costt 7t [22014 it - 4 sintil, 774 5sihat 5 cosat x (t) = 7+ (2 cost- 4 sin4t7 DE 50947 (t)= at [2sinat + 40487 5 sihat

acneral solution of homogeneous problem let) - 43444) + (%(t) che cott-treet sinat 7 50?t cos 4t J hetin +41 taste sett antt ht Fundamental matrix Axt) =(22+ cortt-42t singt 207t sin at the last intttte eitt 57t cott selt sinat I at cos 4t - 4 - Smilt 2 e Smut the cost x"(t) = 2 etort sett sinnt - D ad - b c 1991' ata [ ] ad-bc = [2 et cousut - 4 et sinut] [5 et Smut] - 507 cosut [2 et smut the casual - 14t cunt & cost = - 20 € 14t - - 20 e

Then 1-50 sinat - 7t 2 e sin4x the 't losst LO 20 5 et costit 2 e cott-40 t sintet 7 - 20 7+ . 4 wos4 t 42 sinate A ett sinat deltsinat + 1 e t wat et cost – tett + ett sine tezt [760344 - Asin 4t) 78 4"eft*** 2.17 [5sintt 187tcott 1688 tsing & +32472 tu4+ +16777t sinat 2013 It costot + tot it sinta 15cott Hance general solution of nonhomogeneous problem Jean be given by - X(t) = X(+) Co + Xw/ x'ce) FCE) at co. (a) where, X(t), X(+ f(t) are calculated in ①, ② and @.