Question

find the particular integral of (D^2+16) y= sin4x​

Answer 1

P.I. = 1/(D² + 16) x sinx

= Imaginary part of 1/(D² + 16) x (cosx + i sinx)

= Imaginary part of 1/(D² + 16) x e^ix

= Imaginary part of e^ix . 1/[(D + i)² + 16] x

= Imaginary part of e^ix. 1/[D² + 2iD + i² + 16] x

= Imaginary part of e^ix. 1/[D² + 2iD + 15] x

= Imaginary part of

e^ix. 1/15[1 + 2iD/15 + D²/15]x

= Imaginary part of

e^ix. (1/15). [1 + 2iD + D²] ^-1 x

=Imaginary part of

e^ix. (1/15).[1 - 2iD - D² +…..] x

=Imaginary part of e^ix. (1/15).[x - 2i D x]

= Imaginary part of (1/15). (cosx + i sinx). (x - 2i)

= Imaginary part of

(1/15). (x cosx+2 sinx) + i (x sinx - 2 cosx)

= (1/15). (x sinx - 2 cosx)