1. ## Undetermined coefficient problem

Find a particular solution of $y^{(3)}+y'=2-sinx$

Solution:

The homogeneous solution is $y_h(x)=C_1+C_2 \cos x + C_3 \sin x$

Now, my first guess for the particular solution is: $y_p(x)=A+B \sin x+C \cos x$

I have all terms repeated here, so multiply by x to all, obtains:
$y_p(x)=Ax+B x \sin x + C x \cos x$

Would that be the right guess? Somehow the answer that I obtain from this doesn't match with the one in the book.

Thanks.

You're doing great!

Find a particular solution of: . $y'''+y'\:=\:2-\sin x$

Solution:

The homogeneous solution is: . $y_h(x)\:=\:C_1+C_2 \cos x + C_3 \sin x$ . . . . Right!

Now, my first guess for the particular solution is: . $y_p(x)\:=\:A+B \sin x+C \cos x$

I have all terms repeated here, so multiply by x to all, obtains:
. . $y_p(x)\:=\:Ax + Bx\sin x + Cx\cos x$

Would that be the right guess? . . . . Yes!
You're probably making simple errors here and there.
It took me a few tries to get it right . . .

I finally got: . $A = 2,\;B = \tfrac{1}{2},\;C = 0 \quad\Rightarrow\quad\boxed{ y_p \:=\:2x + \tfrac{1}{2}x\sin x}$