Question

Find constants A and B such that the function y = A sin(x) + B cos(x) satisfies the differential equation y'' + y' − 2y = sin(x).

Answer 1

A = - 0.3 and B = - 0.1.

Explanation:

y(x) = A sinx + B cosx

y'(x) = A cosx - B sinx

y"(x) = -A sinx - B cos x

So, substituting we have:

y" + y' - 2y = -A sinx - B cos x + A cosx - B sinx - 2(A sinx + B cosx)

= -3A sin x - B sinx + A cosx - 3B cos x = sinx

sin x (-3A - B) + cos x(A - 3B) = sin x

Since this is an identity then we equate coefficients:

-3A - B = 1

A - 3B = 0

Solving these we get A = - 0.3 and B = - 0.1.