Second-Order ODE Solved

This problem asks for the general solution of a second-order homogeneous linear differential equation with constant coefficients. The candidate is expected to translate the differential equation into an algebraic condition, recognize the qualitative behavior implied by that algebraic form, and then express the full family of solutions. The setup is a standard textbook-type ODE, the sort that often appears in quant interviews as a proxy for comfort with continuous-time models and linear operators, though it is equally at home in pure calculus or early differential equations courses.

The solution relies on forming and analyzing the characteristic polynomial associated with the differential operator, identifying its roots, and then converting those roots into a real-valued basis of solutions. It leans on exponential ansatz techniques, Euler's formula for handling complex conjugate roots, and the idea that linear combinations of independent solutions span the solution space. Interviewers watch for clean algebra, correct treatment of complex roots, and an understanding of why the resulting functions indeed form the general solution, not just a particular one.

What it tests

Second-order homogeneous linear differential equations with constant coefficients are governed by the behavior of their characteristic equation, a quadratic whose roots dictate the solution's structure. The key insight is that the solution space is spanned by exponential functions whose exponents are the roots of this equation. When the roots are complex conjugates, the real and imaginary parts of the exponentials combine, via Euler's formula, into damped (or growing) oscillatory solutions: exponentials multiplied by sines and cosines. This pattern arises because the differential operator acts linearly, and exponentials are eigenfunctions of differentiation, so the general solution must be a linear combination of such functions. The form of the solution is thus determined entirely by the algebraic nature of the characteristic roots, not by the specific coefficients themselves.