IB MATH AA HL, Paper 3, November, 2021, TZ0, Solved Past Paper

Confirm if the function (u = f (t)) meets the requirements of the differential equation $(\frac{d^{2}u}{d{t}^2}=u)$. This task involves checking whether the second derivative of (u) with respect to (t), notated as $(\frac{d^{2}u}{d{t}^2})$, is equal to the function (u) itself.

Prove that the sum of the squares of two functions (f (t)) and (g(t)) equals the value of the first function evaluated at twice the input, or mathematically, verify that:

((f (t))² + (g(t))² = f (2t)).

This question is asking for a demonstration that a specific relationship holds between two functions and their evaluations at certain points. It involves squaring each of the functions (f ) and (g), adding these squares, and then comparing the result to the value of the function (f ) evaluated at double the initial input (t). The task is to establish this equality under given conditions or assumptions.

In this query, we're utilizing the Euler's formula for complex exponentials, which equates (e^iu) to (cos u + i sin u). The task is to rewrite the formula (e^iu = cos u + i sin u) using solely expressions that involve sin u and cos u. This involves decomposing the exponential function into its trigonometric components.