Linear Equation With Constant Coefficients

Linear Equation with Constant Coefficients:

1. Solve the simultanous equations

Solution:

Using the Operator D = d/dt

(D 2)x 3y = 2e^2t

3x (D 2)y = 0.......(2)

Solving (1) and (2) eliminate (x):

3 × (1) - (2) × (D 2) ⇒ (D² 4D - 5y)y = -6e^2t

A.E: m² 4m - 5 = 0

m = 1,-5

C.F = Ae^t Be^-5t

P.I =

∴ Solution is

2 . Solve given that t = 0, x = 1, y = 0

Solution: Dx y = sin t........(1)

x Dy = cost...............(2)

Eliminate x: (1) - (2)D ⇒ y - D²y = sint sint

C.F = Ae^t Be^-t