Question

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

Answer 1

Integrate both sides twice to recover the general form of $f(x)$:

$f''(x)=5+\cos x\implies f'(x)=5x+\sin x+C_1\implies f(x)=\dfrac52x^2-\cos x+C_1x+C_2$

With the given boundary conditions, we have

$\begin{cases}-1+C_2=-1\\\\\dfrac52\left(\dfrac{3\pi}2\right)^2+\dfrac{3\pi}2C_1+C_2=0\end{cases}\implies C_1=-\dfrac{15\pi}4,C_2=0$

So,

$f(x)=\dfrac52x^2-\cos x-\dfrac{15\pi}4x$