Question

Solve the system. (Please show all work.) I will be rewriting it in operator notation as shown below

$\frac{d^2x}{dt^2}-4x-2\frac{dy}{dt}+y=t \Rightarrow (D^2-4)x+(1-2D)y=t$

$2\frac{dx}{dt}+x+\frac{d^2y}{dt^2}=0 \Rightarrow (2D+1)x+D^2y=0$

Answer 1

Ans Given that Equation can be re-written as (0²-4) x + (1-2D) y = t where Doa Equation can be re-written as (1 +20)x + 2y = 0 eq. & xD2 – ca. 4 x (1-2), we get D? (24)x - (1+20) (1-2D)ze = ? (+) (84-4D²)* - (1-41?) x = d 4 : De la (04-482-1 + 492)x = 0 (+) = 1 + C4)-o (07-1)x=0 .: Auxiliary equation will be mt-1=0

(m²-1) (m2+1) = 0 => m2-1-o , m?+1 to > m=# , mrti Complementary function will be C.F. = Get + Get & Cz Cost + 4 sint where G, C2, C3, C4 are any arbitrary Constants. HOW : Right hand side of ea. 6 is zero - particular integral (P.I.) = 0 This solution to eq. 5 will be x = C.F. & P. I. x = Get + Cet + C2 Cast & Ca sint to x = Get + (zet + Cg cott & Ca sint How eq. x (1 +2D) – eq. Ax (6²4), we get (1 +2D) (1-20) y - 02 (024) y = (1 +20)* (1-4D²) y - (bt - 402) y = t +2 (1-45² - Dt + 48²) y = x + 2 (1-DA) y = x+2 - equation will be 1-mt = 0 - (1-m²) 114m²) = 0 1-m²=0 , 1 + m² = 0 = m = I1 & mati Complementary function will be CoF = Get +get + (z Cast & Casint where G, C2, C3, CA are arbitrary Constants.

Mow particular integral (P.I.) = 1 A+ 2) (1-DA) - P.J. = (1-DA) (t + 2) = (1 + Dt + Do + ...) (t +2) = (t + 2) +0 = A +2 P.I. Thus solution to ea. 6 will be Y = CF. & P.I. > y =Get + let + C₂ Cast & C sind + t +2 Hence solution to the given system of differential equation is x = Get + Cet & Cz Cost + C sint y = Get + C et + C₂ Cast + 4 fint + A +2