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

To determine which statements about the periodicity of solutions to the differential equation \(\frac{d^2y}{dx^2} + 4y = \sin(ax)\) are true, let's analyze the problem step-by-step. 

Understanding Periodicity: A function \(y(x)\) is periodic if there exists a real number \(T > 0\) such that \(y(x + T) = y(x)\) for all \(x \in \mathbb{R}\). Thus, the periodicity of solutions to the given differential equation depends on the form of the solution and the properties of the driving term, \(\sin(ax)\).

Analyzing the Differential Equation: The differential equation \(\frac{d^2y}{dx^2} + 4y = \sin(ax)\) is a non-homogeneous linear differential equation with a sinusoidal driving term. The general solution is the sum of the complementary solution (general solution of the homogeneous equation) and a particular solution of the non-homogeneous equation.

\[y_c'' + 4y_c = 0\]

The complementary solution \(y_c\) can be expressed as: 

\[y_c(x) = c_1 \cos(2x) + c_2 \sin(2x)\]

where \(c_1\) and \(c_2\) are arbitrary constants.

Particular Solution: A particular solution \(y_p(x)\) can be found using the method of undetermined coefficients. Assume: 

\[y_p(x) = A \sin(ax) + B \cos(ax)\]

so 

\[y_p''(x) = -Aa^2 \sin(ax) - Ba^2 \cos(ax)\]

Substituting \(y_p\) and its derivatives into the differential equation gives conditions for coefficients \(A\) and \(B\).

Conditions for Periodicity: The overall solution is given by: 

\[y(x) = c_1 \cos(2x) + c_2 \sin(2x) + A \sin(ax) + B \cos(ax)\]

For \(y(x)\) to be periodic, the frequency components in \(2x\) and \(ax\) must be commensurable. This occurs when \(a\) is a rational number except \( -2 \) or \( 2 \). If \(a\) is irrational, the sinusoidal components \(\sin(ax)\) and \(\cos(ax)\) do not repeat, resulting in non-periodic behavior, except for trivial cases where the periodic component vanishes.

Conclusion: Based on the analysis, we conclude the following:

• All solutions of \((*)\) are periodic for every choice of \(a \in \mathbb{Q} \setminus \{-2, 2\}\).
• If \(a \in \mathbb{R} \setminus \mathbb{Q}\), there is a unique periodic solution (trivial case), indicating non-permeance of other solutions being periodic.