Find f. f ''(x) = 5 + cos(x), f(0) = −1, f(3π/2) = 0

The required function is .

Given:

The given differential equation is:

The initial values are .

To find:

The function .

Explanation:

We have,

Integrate both sides with respect to .

Integrate both sides with respect to .

        ...(i)

We have . So,

We have . So,

On further simplification, we get

Substituting , we get

Therefore, the required function is .

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Integrate both sides twice to recover the general form of :

With the given boundary conditions, we have

So,