Question

A real-valued function y(x) defined on $\R$ is said to be periodic if there exists a real number $T\gt0$ such that y(x + T) = y(x) for all $x\isin\R$. Consider the differential equation
$\frac{d^2y}{dx^2}+4y=\sin(ax), x\isin\R, \quad (*)$
where $a\isin\R$ is a constant.
Then which of the following is/are true?

• A. All solutions of $(*)$ are periodic for every choice of a.
• B. All solutions of $(*)$ are periodic for every choice of $a\isin\R$- {-2, 2}.
• C. All solutions of $(*)$ are periodic for every choice of $a\isin\mathbb{Q}$-{-2, 2}.
• D. If $a\isin\R-\mathbb{Q}$, then there is a unique periodic solution of $(*)$.

Answer 1

Answer: (C), (D)

All solutions of $(*)$ are periodic for every choice of $a\isin\mathbb{Q}$-{-2, 2}.

Answer 2

The correct option is (C): All solutions of $(*)$ are periodic for every choice of $a\isin\mathbb{Q}$-{-2, 2}. and (D): If $a\isin\R-\mathbb{Q}$, then there is a unique periodic solution of $(*)$.