Question

Solve the non-homogeneous IVP of the systems of equations showing all details

y’₁ + 3 y₁ +4 y₂ – 5 exp (t) = 0

y’₂ – 5 y₁ - 6y₂ + 6 exp (t) = 0

y₁(0) = 19    y₂(0)= -23

Answer 1

y1' +3y1 +4y2 - 5exp(t) = 0

Taking laplace, with y1(0)= 19

sy1(s) - y1(0) + 3y1(s) + 4y2(s) - 5/(s-1) = 0

(s+3)y1(s) - 19+ 4y2(s) - 5/(s-1) = 0 ......(1)

y2'- 5y1 -6y2 +6exp(t) = 0

Taking laplace with y2(0) = -23

sy2(s) - y2(0) - 5y1(s) - 6y2(s) + 6/(s-1) = 0

(s-6)y2(s) +23 -5y1(s) + 6/(s-1) = 0......(2)

From eq(1),

y1(s) = 5/(s-1)(s+3) +19/(s+3) -4y2(s)/(s+3) put into eq(2)

(s-6)y2(s) +23 -25/(s-1)(s+3) -85/(s+3) -20y2(s)/(s+3) + 6/(s-1) = 0

y2(s) * {(s-6) -20/(s+3)} = 25/(s-1)(s+3) -6/(s-1) +85/(s+3) -23

y2(s) (s²-3s -38)/(s+3) = 25/(s-1)(s+3) -6/(s-1) + 85/(s+3) -23

y2(s) = 25/(s-1)(s²-3s-38) -6(s+3)/(s-1)(s²-3s-38) + 85/(s²-3s-38) -23(s+3)/(s²-3s-38)