Question

(D³-5D²+7D-3)y=e²coshx

Answer 1

 y = e² cosh x = e² [ e^x + e^-x ] = e^2x + e^0  = e^2x  + 1

D = dy/dx = 2 e^2x
D² = d² y / dx² = d/ dx  (dy/dx) = 4 e^2x
D³ = d³ y / dx³  = d/dx (d²y /dx)  = 8 e^2x
D³-5D² +7 D - 3  = 8 e^2x - 5*4 e^2x + 7 * 2 e^2x - 3