## 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) ⇒ (D^{2} 4D - 5y)y = -6e^{2t}

A.E: m^{2} 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 - D^{2}y = sint sint

C.F = Ae^{t } Be^{-t}