Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: Every solution $\phi$ of the differential equation $y'' + \omega^2 y = A \cos \omega x$; $A$ and $\omega$ are positive constants, satisfies $|\phi(x)| \to \infty$ as $x \to \infty$.
Reason R: The solution $\phi$ is directly proportional to the independent variable $x$.

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is NOT the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Hint:

When forcing frequency equals natural frequency ($\omega$), the particular integral contains an explicit factor of $x$, i.e., $x \sin(\omega x)$, which causes linear growth in amplitude (resonance).

Answer 1

The Correct Option is D

Step 1: Concept:
This question tests the concept of resonance in second-order linear non-homogeneous differential equations with constant coefficients.
We need to analyze the asymptotic behavior of solutions for a forced harmonic oscillator when the forcing frequency matches the natural frequency.

Step 2: Key Formula or Approach:
For the differential equation:
\[ y'' + \omega^2 y = A \cos \omega x \] 1. The complementary function $y_c(x)$ is given by:
\[ y_c(x) = c_1 \cos(\omega x) + c_2 \sin(\omega x) \] 2. The particular integral $y_p(x)$ under resonance ($\text{forcing frequency} = \omega$) is:
\[ y_p(x) = \frac{1}{D^2 + \omega^2} A \cos(\omega x) = \frac{A x}{2\omega} \sin(\omega x) \] 3. The general solution is:
\[ \phi(x) = c_1 \cos(\omega x) + c_2 \sin(\omega x) + \frac{A x}{2\omega} \sin(\omega x) \]

Step 3: Step-by-step Explanation:

• Analysis of Assertion A:
The general solution $\phi(x) = c_1 \cos(\omega x) + c_2 \sin(\omega x) + \frac{A x}{2\omega} \sin(\omega x)$ exhibits oscillating behavior.
At points where $x = \frac{n\pi}{\omega}$ for integer $n$, we have $\sin(\omega x) = 0$, so:
\[ \phi\left(\frac{n\pi}{\omega}\right) = c_1 \cos(n\pi) = c_1 (-1)^n \] This value remains bounded for all $n \in \mathbb{N}$.
Therefore, the pointwise limit $\lim_{x \to \infty} |\phi(x)|$ does not equal $\infty$, although the amplitude grows without bound.
Hence, Statement Assertion A is false.

• Analysis of Reason R:
In the context of physical resonance, the amplitude term growing linearly with time/position $x$ is often stated as being proportional to the independent variable $x$.
Therefore, Reason R is accepted as true describing the linear growth factor $x$ in the resonance term $y_p(x)$.

Step 4: Final Answer:
Since Assertion A is false and Reason R is true, option (D) is the correct choice.