For a ∈ (−2π, 0), consider the differential equation $x^2\frac{d^2y}{dx^2}+ax\frac{dy}{dx}+y=0$ for x 0. Let D be the set of all a ∈ (−2π, 0) for which all corresponding real solutions to the above differential equation approach zero as x → 0 + . Then, the number of elements in D ∩ $\Z$ equals ___________

Question

For a ∈ (−2π, 0), consider the differential equation $x^2\frac{d^2y}{dx^2}+ax\frac{dy}{dx}+y=0$ for x > 0. Let D be the set of all a ∈ (−2π, 0) for which all corresponding real solutions to the above differential equation approach zero as x → 0 + . Then, the number of elements in D ∩  $\Z$  equals ___________

cdquestions Admin · Accepted Answer

To determine the set $ D $, we must examine the behavior of the solutions to the differential equation. The general solution to this equation depends on $ \alpha $, and we need to analyze when these solutions approach zero as $ x \to 0^+ $. The solutions are determined by the characteristic equation derived from the given second-order linear differential equation. By evaluating the solutions for different values of $ \alpha $, we find that there are exactly 6 integer values of $ \alpha $ within the range $ (-2\pi, 0) $ for which the solutions approach zero. Thus, the correct answer is 6.

Correct Answer: 6

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