Write the given differential equation in the formL(y) = g(x),where L is a linear differential operator with constant coefficients. If possible, factor L. (Use D for the differential operator.)y'' − 3y' − 28y = x − 2

L(y) = g(x)

Step-by-step explanation:

Step(i):-

Given differential equation

  y'' − 3 y' − 28 y = x − 2

Operator form ( D² - 3 D - 28 )y = x -2

                         f( D ) y = Ф(x)

Auxiliary equation

                    f( m ) = m² - 3 m - 28

          ⇒  m² - 3 m - 28 =0

         ⇒  m² - 7 m +4 m - 28 =0

         ⇒ m ( m -7 ) + 4 ( m -7) =0

         ⇒ (m + 4)( m -7) =0

       ⇒ m = -4 and m =7

Complementary function

Step(ii):-

Particular integral

( 1 + x )⁻¹ = 1 - x + x² - x³ +.....

  we use notation

Higher degree terms will be neglected

        D(x) =1

        D²(x) =0

  The particular integral

Conclusion:-

General solution of given differential equation

step-by-step explanation: