Question

1) Solve the following second order linear ODEs by i. Finding the roots of differential operator on the Left Hand Side; ii. Finding the roots of the annihilator of the Right Hand Side of the equation; If the Right Hand Side is zero, there is nothing to do here! iii. Writing down a basis of the solution of the homogeneous equation; iv. Reducing the task of finding the undetermined coefficient to solving a system of linear equations in augmented matrix form by finding the column vector v for the Right Hand Side in terms of the basis and the matrix of differentiation A in terms of the basis. HINT: Ann 6.")=(0-2): Roots = 2, m +1 times; m! HINT: Ann Fes" {cos(Bt), cos(pt)} = ((D-(a+Bi))(D-(a–Bi): m! Roots ee"{cos(Bt),cos( Bt)} = a +bi, m+1 times; c) (D-(7))'(D-(8)° = 7te" cos(51); d) ((D-(7+8i))(D-(7-8i))) = 0; e)((D-(7+8i))(D-(7-8i)))* = 71*e"; A) ((D-(7+8i))(D-(7-8))) = 7te” cos(5t).

I need help solving the second-order linear ordinary differential equations for part c d e and f. Please show all work thank you

Can you please answer part i ii iii and iv for part c and the same i through iv for part d and so on for parts e and f.

C and d

Answer 1

1 (i) finding (ii) Finding LHS equation. the root of differential operater on the the roots of the Annihilator of RHS of Hu (ih) writing down of the solution of the homogeneous equation (c) (0-7) 3 (0-8) 2 = 7 te 2 tcas (5t) (m-7) 3 (m-8)2 = 0 m=7,7,7,8,8 (C, TC ₂ U + C₂ x 2) etx - (L) Ite et cos (st) Auxiliary equation (C4+ (9x) x 8 x Annihilater Annihilator of any e ax cospx. where n=2 xn-ear [[p-)+BP3" a=2 ß = 5 1 (0-2) + 231 = (²+4-40+25)? = (0²4D +29)² – (ii) Particular integral P. 7e2t cos (St) (D-7)3(0-8) t cos (5+) (0 +2-7)3 (0+2-8) tcas (st) (0-5)3 (D-672 2 7 e27 7 ezt S securex

20 76 Teu i cos(st) Co-5)3(p on t cos (56) (15)(+25 POD) (D'+36 +2D) Tert cos (st) (0-5) (-25+25-10D) (-25+36 +12D) t t D'DP- 7 et cosst t -1010²-5D) (12D+9) 7 et casst D(0²_5) (4013) - 7 eet casst I IZ 30 4D², 3D-201-15 - 7 e 2 casst 4D²_17D-15 D +2 30 Foxit censt 3) - prapy) (1 cosst (t² + 40² 4 2 4 17 pt² e 21 cosst (40²-17D e2 30xis 7 e24 T5 15 30x15 e et lasst ( +² + 8 1 4 1742 + I P.I 15 30XIS S securex

(d) (10-(7 +Bi)) (0-17-8;)))2 = 0 Auxiliony equation ((M-17+ 8;) (m-(7-8:'))) ? (ni-(7+81))2 (M-(7-8))2,0 •cy m = 7+bi, tebi; 78;,7 8i C.F. (a + C₂x) x ( C3 (C3 Ces&xt sin 8x). store A