Linear Equation With Constant Coefficients

Linear Equation with Constant Coefficients-1:

3. Solve

Solution: Dx 2y = -sint..............(1)

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

1×2 (2) × D ⇒ 4y D²y = -2sint D(cost)

                    ⇒ (D² 4) = -3 sint

                     m² 4 = 0, m = ±i2

             C.F = A cos 2t B sin 2t

             P.I =

             y = A cos 2t B sin 2t - sint

   ∴            x = A cos 2t B sin 2t - cost

                 y = A cos 2t B sin 2t - sint.

4. Solve

Solution: Dx (-D 2)y = cos 2t......(1)

(D - 2)x Dy = sin 2t......(2)

Eliminating y from (1) and (2)

(D² - 2D 2)x = -2 sin 2t cos 2t

R.H.S = 0 ⇒ m² - 2m 2 = 0

m = 1±i