Find the particular solution to y " = 3sin(x) given the general solution y = -3sin(x) + Ax + B and the initial conditions y(pi/2)=-pi and y'(pi/2)=-2

Question

Find the particular solution to y " = 3sin(x) given the general solution y = -3sin(x) + Ax + B and the initial conditions  and .

A. -3sin(x) + 2x + 3

B. -3sin(x) + pi

C. -3sin(x) - 2x + 2pi + 3

D.-3sin(x) - 2x + pi

Answer 1

Since y" = 3sinx, y' = 3∫sinxdx = -3cosx + C

Since y'(π/2) = -2, we have -3(0) + C = -2. So, C = -2.

y' = -3cosx - 2

y = ∫(-3cosx - 2)dx = -3sinx - 2x + C₁

Since y(π/2) = -π, -3(1) - 2(π/2) + C₁ = -π

So, C₁ = 3

Therefore, y = -3sinx - 2x + 3