Find the most general function f such that
f′′(x) = 16 cos 4x .
To find the most general function f that satisfies the given second-order differential equation f''(x) = 16 cos 4x, we will integrate the equation twice.
Step 1: Integrate the equation f''(x) = 16 cos 4x with respect to x to find the first antiderivative.
∫ f''(x) dx = ∫ 16 cos 4x dx
By integrating the right side, we get:
f'(x) = 16/4 ∫ cos 4x dx
f'(x) = 4 ∫ cos 4x dx
Using the integral identity for the cosine function, we have:
f'(x) = 4 (1/4) sin 4x + C1
f'(x) = sin 4x + C1
Here, C1 is an arbitrary constant.
Step 2: Integrate the equation f'(x) = sin 4x + C1 with respect to x to find the second antiderivative.
∫ f'(x) dx = ∫ (sin 4x + C1) dx
By integrating the right side, we get:
f(x) = ∫ sin 4x dx + ∫ C1 dx
f(x) = -1/4 cos 4x + C1x + C2
Here, C2 is an arbitrary constant.
Therefore, the most general function f that satisfies the given differential equation f''(x) = 16 cos 4x is:
f(x) = -1/4 cos 4x + C1x + C2