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

The Correct Option is C, D

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 \((*)\).