# Solve the given system of equations by systematic elimination method ;

DX+Z=e^t

(D-1)x+Dy+Dz =0

x+2y+Dz=e^t

Solution;

First of all write equations in such a way to find X(t),Y(t),Z(t)

X' + Z = e^t or Z = e^t - X' -----(1)

X' - X + Y' + Z' = 0 ------(2)

X+2Y+Z' = e^t -------(3)

Now find Z'

Z' = e^t - X" ------(4)

Now put Z' in (2) and (3)

X' - X + Y' + e^t - X" = 0 ------(5)

X" - X' + X - Y' - e^t = 0 -------(5)

X + 2Y + e^t - X" = e^t here

Y = X" - X / 2 ----(6)

Y' = X"' - X' / 2 -----(7)

Put equation (7) in (5)

X" - X' + X - (X"' - X')/2 - e^t = 0

X"' - 2X" + X' - 2X = - 2e^t -----(8)

Equation (8) is non homogeneous equations having solution

X = Xc + Xp

X(t) = C1e^2t + C2Cost + C3Sint + e^t ---(9)

X' = 2C1e^2t - C2Sint + C3Cost + e^t

X" = 4C1e^2t - C2Cost - C3Sint + e^t

Put X , X' and X" in (6) and (1) respectively to find

Y(t)=3/2 C1e^2t - C2Cost - C3Sint

Z(t)= −2C1e^2t − C3cost + C2Sint