The first thing we must do in this case is find the derivatives: y = a sin (x) + b cos (x) y '= a cos (x) - b sin (x) y '' = -a sin (x) - b cos (x) Substituting the values: (-a sin (x) - b cos (x)) + (a cos (x) - b sin (x)) - 7 (a sin (x) + b cos (x)) = sin (x) We rewrite: (-a sin (x) - b cos (x)) + (a cos (x) - b sin (x)) - 7 (a sin (x) + b cos (x)) = sin (x) sin (x) * (- a-b-7a) + cos (x) * (- b + a-7b) = sin (x) sin (x) * (- b-8a) + cos (x) * (a-8b) = sin (x) From here we get the system: -b-8a = 1 a-8b = 0 Whose solution is: a = -8 / 65 b = -1 / 65 Answer: constants a and b are: a = -8 / 65 b = -1 / 65
