Second Order Differential Equation with Initial Conditions

Solve the second order constant coefficient differential equation $y'' - y' - 6y = 0$ with initial conditions $y(0) = 1$ and $y'(0) = 2$.

This problem addresses a fundamental concept in differential equations: solving a second order linear homogeneous differential equation with constant coefficients and given initial conditions. These types of equations are typical in modeling natural phenomena and engineering systems, such as mechanical vibrations and circuit analysis. To solve this problem, one must understand the characteristic equation, which arises by assuming solutions of a specific exponential form. The nature of the roots of this characteristic equation, whether they are real and distinct, real and repeated, or complex, will determine the form of the general solution. Additionally, it's essential to apply initial conditions to solve for any arbitrary constants within the general solution. This technique showcases how initial conditions are crucial in determining a unique solution that accurately models the system at particular starting points. Overall, solving such differential equations develops competencies in mathematical modeling and analytical problem-solving, which are indispensable in advanced engineering and physics courses.