Question

Find the particular integral of (D²+9)y = cos3x.

Answer 1

 (D² + 9)y = sin3x

It's auxiliarly equation is 

m² + 9 = 0

⇒ m = \(\pm\) 3c

\(\therefore\) C.F. = c₁ cos3x + c₂ sin3x

P.I. = \(\frac1{D^2 + 9} sin3x\)

\( = \frac{-x}{2\times3} cos3x\)         \(\left(\because \frac1{D^2 + a^2}sin\,ax = \frac{-x\,cos \,ax}{2a}\right)\) 

\( = \frac{-x}{6}cos3x\)

Particular integral = \( \frac{-x}{6}cos3x\)