Question

(a) Solve the initial value problem 2" +2r' + r = 8(t - 2), z(0)=1, 2'0) = 2 (b) Consider the initial value problem -2 -5 z(0) = 3 Find ö(t), writing your answer as a single vector. k 2 k 0 1] (c) Consider the matrix 0 -2 k 3 i. Compute the determinant. ii. For what value(s) of k does A exist? iii. For what value(s) of k does the linear system A7 = 7 have nontrivial solutions? iv. For what value(s) of k does A have zero as an eigenvalue? v. For any vector be R*, find the value(s) of k for which the linear system A7 = b has a unique solution.

Answer 1

Aven (4-2)$ = = a + اه را= (6) X (= (ه) * on both sides + (10w408 مقهاها Apply (2-t) } ا =143 + 213 + { }L و (xb 3 - 433اوه + (۵) - (۵) × 5 - {داء د که : {۶}ا + 2+25 14x3 - 2 + L] è as 5 - s2473 د 13 (1 + 5 + ی) + + كے - ~ }ا 35+s+ - + St 4 (+1) 2 ۱+ و +ی 25. = 3 ما 5 + (+1) 2 ( 5+1) (511) 2 ve footos معهpها APP9 ,هلا obtian و از (4)= کتا دل ہوا + + .} لے (511) (2+1) (5+1)² Now ودمان دے (+1) { روی کا tét (5+1)

u(t-2) و2 م كات ( (t-2) 8 (4-2) NOW IS ı (3+1)? i'{ csovs ] { tét \"{ stij}a te t"{ is ve} = (174) }- cop tët) - - ēt (t-1) .. & finally "{ wat e Hoe ( fe et (t-1) +utet 4(1-2) X(t) = (t-2) ē (4-2) (+-2) ét. é uct-2) - ē (t-1)+4tēt (t) = et X(t) = * (é(t-2) ult-2) -3t +1) alt) = ét (é? (t-2)uct-2) – 3t+1) 07 post other questions seperately