Question

D²-3d+2)y=cos3x.cos2x

Answer 1

(D² - 3D + 2) y  = cos 3x cos 2x

= 1/2(cos(3x + 2x)+ cos (3x - 2x))

= 1/2 (cos 5x + cos x)

Auxiliary equation is

m² - 3m + 2 = 0

⇒ (m - 2) (m - 1) = 0

⇒ m = 1 or m = 2

C. F. = c₁e^x + c₂e^2x

P. I. = \(\frac{1}{D^2-3D+2}\frac{1}2(cos\,5x + cos\,x)\) 

= \(\frac12\left(\frac{1}{D^2-3D+2}cos\,5x+\frac{1}{D^2-3D+2}cos\,x\right)\) 

= \(\frac12\left(\frac{1}{-25-3D+2}cos\,5x+\frac{1}{-1-3D+2}cos\,x\right)\) 

= \(\frac12\left(\frac{-1}{3D+23}cos\,5x+\frac{1}{1-3D}cos\,x\right)\) 

= \(\frac12\left(\frac{-1(3D-23)}{9D^2-529}cos\,5x+\frac{1+3D}{1-9D^2}cos\,x\right)\) 

= \(\frac12\left(\frac{(3D-23)}{-9\times25-529}cos\,5x+\frac{1+3D}{1-9x-1}cos\,x\right)\) 

= \(\frac12\left(\frac{(3Dcos\,5x-23cos\,5x)}{754}+\frac{cos\,x+3Dcos\,x}{10}cos\,x\right)\)

= \(\frac12\left(\frac{-15sin\,5x-23cos\,5x)}{754}+\frac{cos\,x-3sin\,x}{10}\right)\) 

Complete solution is

y = C. F. + P. I.

= c₁e^x + c₂e^2x - \(\frac12\left(\frac{15sin\,5x+23cos\,5x)}{754}+\frac{3sin\,x-cos\,x}{10}\right)\)