The indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given non-homogeneous equation. y'' − 5y' + 4y = x; y1 = ex
Find yp(x).

The particular integral of given differential equation

General solution of given differential equation

Step-by-step explanation:

Step(i):-

Given Differential equation  y'' − 5 y' + 4 y = x

Given equation in operator form

        D²y - 5 Dy +  4 y = x

⇒     ( D² - 5 D +  4 ) y =x

⇒    f(D) y = Q

where  f(D) = D² - 5 D +  4 and Q(x) = x

The auxiliary equation  f(m) =0

          m²-5 m + 4 =0

         m² - 4 m - m + 4 =0

        m ( m -4 ) -1 ( m-4) =0

         (m - 1) =0   and ( m-4) =0

        m = 1 and m =4

The complementary function

Step(ii):-

particular integral

Particular integral

taking common '4'

applying binomial expression

     ( 1 + x )⁻¹    = 1 - x + x² - x³ +.....      

Now simplifying and we will use notation D =

Higher degree terms are neglected

The particular integral of given differential equation

Final answer:-

         General solution of given differential equation

hiking down a winding trail.

step-by-step explanation: