Special Functions

The given differential equation is: This is a form of Hermite's differential equation, which is satisfied by the Hermite polynomials for integer values of . The general solution of this equation is: where is the n-th Hermite polynomial.
The function is given by: The behavior of depends on the values of . 1. For : The solution is , and thus . This corresponds to a Gaussian curve, which is symmetric and bell-shaped. This corresponds to option (B) in the graph, where the plot for is a smooth Gaussian-like curve. 2. For : The solution is , and thus . This corresponds to a curve that starts at 0, increases, and then decreases as increases. It also shows an oscillatory nature due to the presence of the term multiplied by the exponential term. This corresponds to option (C) in the graph, where the plot for is an oscillatory curve with one zero crossing. Thus, the correct answer is option (B) for and option (C) for .