Question

Solve the following first order linear non-homogeneous system using projection matrices: s x' = 2x – y + 1, x(0) = 0 ly' = 3x – 2y + 2, y(0) = 2 where x = x(t), y = y(t).

Answer 1

page- Answer - Given data ge'= an-yt', 2400) =0 y'= 34-ay ta, y(0) = 2 where u= a(4), y = y(t) we have to using the laplaco tounfam obs geven Equations lucm) = 2() - Lly) +L(1) & Chature -(an - (N) 78 - Chootinje -a- (R7S |(S-a) L(u)+(cy)= $10 Similarley we falce the Second Equalion y'=31-34+2,400) = 2 apply the Laplace fan sforma LIGJ = 34(u) -al ty) ta to --3LCD)+ SL (y) – 4(0) + 22(y) = -3L(1) + Se ly) -o taly) = 5 1-34(x) + ($+2) L(Y) = 3+2 -☺

page/-6 ③ to calculate We have to take the equation ② and Now we take the equation @ and multiple the estaj ((S-2)lex) tecy) > px (S+2) a 6-2) (S+2) L(N) +(8+2) Lly) - 642 43 from the Equation & @ (S-2) CS+2) L(1)+ CS+2){ty) - 642 - 3 LCW) + (Steylly) = atas (8?_4+3)L(x) = 8+4-2-as (531) Lew - Smas 18=1) L(H) = 18 ks) L(W) - on subtraction

page: - we have to take the inverse taplace hanform 1(4)- (ett-et) = 1514) -- Sint a(t) = et et put value of ew-[sti ] in equation @ (20) + Data ent]+1994 dalys - - V - 3+ y) = ^ [a bi + (14) = $ Lly) = + [ 5] Takking enverse laplace hanfoom.. yet - 12 [-3.ettet] YE) = 1+ Bettet

Pagel-ly Hence 26) = fet-fet y (€) = lt met het o- -