Find the particular solution of the differential equation that satisfies the initial condition(s). f ''(x) = sin(x), f '(0) = 8, f(0) = 3

Answer 1

f''(x) = sin(x)

f'(x) = -cos(x) + a

f'(0) = 8 ⇒ -cos(0) + a = 8

-1 + a = 8 ⇒ a = 9

f'(x) = -cos(x) + 9

f(x) = -sin(x) + 9x + b

f(0) = 3 ⇒ -sin(0) + 9(0) + b = 3

b = 3

f(x) = -sin(x) +9x + 3

Test the solution.