Answer:
L(y) = g(x)

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

