Question

Function F satisfies f’(x)=2cosx-3sinx and f(0)=5. Find an expression for f(x).

Answer 1

f(x) = 2sin(x) +3cos(x) +2

Explanation:

You want the function f(x) that satisfies f’(x)=2cosx-3sinx and f(0)=5.

Solution

This differential equation is solved by direct integration.

f(x) = ∫f'(x)·dx

f(x) = ∫(2cos(x) -3sin(x))dx = 2sin(x) +3cos(x) +C

To make f(0) = 5, we must have C = 2:

f(x) = 2sin(x) +3cos(x) +2