Question

F”(theta)= sin(theta) + cos (theta), f(0)=5 , f’(0)=2. Find f.

Answer 1

`f''(\theta)=\sin\theta+\cos\theta`

Integrate both sides:

`f'(\theta)=-\cos\theta+\sin\theta+C_1`

Integrate again:

`f(\theta)=-\sin\theta-\cos\theta+C_1\theta+C_2`

With the given initial conditions, we find

`f(0)=5\implies -1+C_2=5\implies C_2=6`

`f'(0)=2\implies -1+C_1=2\implies C_1=3`

Then

`f(\theta)=-\sin\theta-\cos\theta+3\theta+6`