Question

3. x"(t) - y'(t) = -2t x'(t) + y"(t) = 0 x(O) = 0; y(0)=0; x'0) = 0; y'(0) = 0

Please show all work.

Answer 1

(3) x H) - 1 (4) 1 - 2 x'lt) + y "14) = 0 - () . _ (ii) x 10) = 0, 710)=0, x'(0)=0, 7(0) = 0. taking laplace toansform of equation (A) both sides we get SPX(5) - 52(0) -x0) – (5813) -410)) = SX(S)-x10) + S²Y(5) - 7(0)-710) => . 52 s’x(s) - sy(s) = 5x (5) +s?y(s) - 0 () ou since, 5x (5) + 52415) = 0 sx(5) = -524 (5) x (s) = -5/15)

from equation (in) we get -537/5) - Sy(s) = - 2 / 2 5 (52+1) 715) = Cs) = 31541) By the partial fraction methods A52/52+1) + B5 (5²+1) + c(5²+1) + (ost) 5 = 2 (A + D) st + ( B+ E) + (A+C) 5²+ (B)s + C = 2. yeting equating coefficients coefficients both both sides of sos S2, S3 we get At D = 0 o BtE=0, Atc=0, B = 0 BLE=0, at =2 A = -2, E = 0, D = 2

Hence, 769) = 2 + + toans foron both sides laplace taking inverse we we get a 4 Lt) = -2i+t2 + acost y'lt) = 2t - 2 sint y"(t) = 2 - 2 cost from equation (1) x'lt) to y'lt) = 0 x'ct) = - yult) = -2t 2cost x'lt) = – 2 + acost alt) = -2t ta sint (integrating both sides w.rott)

Therefore, xit) = -2t + sint ylt) = -2+ + 2 + 2 cost