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