Question

Let \( k, \ell \in \mathbb{R} \) be such that every solution of \[ \frac{d^2y}{dx^2} + 2k \frac{dy}{dx} + \ell y = 0 \] satisfies \( \lim_{x \to \infty} y(x) = 0 \). Then which of the following is/are TRUE?

• A. \( 3k^2 + \ell<0 \) and \( k>0 \)
• B. \( k^2 + \ell>0 \) and \( k<0 \)
• C. \( k^2 - \ell \leq 0 \) and \( k>0 \)
• D. \( k^2 - \ell>0 \), \( k>0 \), and \( \ell>0 \)

Hint:

For second-order linear differential equations, the solutions depend on the discriminant of the characteristic equation. If the real part of the roots is negative, the solution decays to zero.

Answer 1

The Correct Option is C, D

Step 1: Analyzing the second-order differential equation.
The characteristic equation associated with the differential equation is: \[ r^2 + 2kr + \ell = 0. \] The solutions to this equation are: \[ r = \frac{-2k \pm \sqrt{4k^2 - 4\ell}}{2} = -k \pm \sqrt{k^2 - \ell}. \]
Step 2: Conditions for solutions to tend to 0.
For \( y(x) \) to tend to 0 as \( x \to \infty \), the real part of the roots must be negative. Thus, we need \( k>0 \) and \( k^2 - \ell \leq 0 \), which ensures the real parts of the roots are non-positive.
Step 3: Conclusion.
Thus, the correct answers are (C) and (D).