y"-y'=sin x

Question

solve in undetermined coefficients

Answer 1

probably sinusoidal

y = a sin x + b cos x
y' = a cos x - b sin x
y" = -a sin x - b cos x = -y

-a sx -b cx -a cx + b sx = 1 sx
b = -a
-a + b = 1

-2a = 1
a = -1/2
b = +1/2
so I get y = -(1/2) (sin x - cos x)

Answer 2

That pesky y' gets in the way. The solution is really

y = c1 e^x + c2 - 1/2 (sinx - cosx)

Answer 3

Whoops, yes, use Steve's