Question

(D³ - 6 D²D' + 11 DD'² + 2 D'³) z=e^5x+6y​

Answer 1

The answer is slightly different but the method is same.

(D^3-6 D^2+11D-6)y=e^-2x

Let f(D)= D^3 -6 D^2+11D--6

The A E is f(m) =0

m^3--6m^2+11m —6 (2)

(m-1) is a factor

( m-2)ia factor

(m-3) is afactor

The Roots of( 2) are 1 2 and 3

CF = yc =c1 e^x + c2 e^2x +c3 e-^3x

yp=PI = 1/ (D-1) (D-2)( D-3) e^-3x

Here f(2)=0 let phi (D) =D-1 then Ph(i2)= 2-1 =/ 0 AND K=2