What is the function?

Question

Find the function F that satisfies the following differential equations and initial conditions.

F''(x)= cosx, F'(0)= 9, F(pi)= 8

Answer 1

F''(x) = cos x // given

F'(x) = sin x + C₁ // by integration

F(x) = -cos x + C₁x + C₂ // by integration

F'(0) = 9 // given

sin 0 + C₁ = 9 // after evaluating the previously calculated F'(0)

C₁ = 9 // solving for C₁

F(π) = 8 // given

-cosπ + 9π + C₂ = 8 // after evaluating the previously calculated F(π)

C₂ = 7 - 9π // solving for C2

F(x) = -cosx + 9x + 7 - 9π // final result