Solving Third Order Differential Equation

Solve the differential equation $y''' + y' = 0$.

This problem involves solving a third order linear differential equation. Differential equations of this type are often solved by finding the characteristic equation, which provides insights into the nature of the solutions. In this case, the differential equation includes both a third derivative and a first derivative, but no zero or second derivative terms. This juxtaposition causes the interaction of the derivatives to determine the behavior and general solution of the equation.

When approaching such equations, a common technique is to assume a solution of a specific form, often involving exponential functions due to their unique properties regarding differentiation. Exponential solutions allow us to transform the differential equation into an algebraic equation, simplifying the process. This characteristic equation can then be solved using algebraic methods to identify the roots, which in turn help formulate the general solution.

Understanding and solving higher order linear differential equations is essential in various fields such as physics and engineering, as they often model complex dynamic systems. By mastering techniques like the characteristic method, you equip yourself with a powerful toolset to solve not only mathematical problems, but also to analyze real-world phenomena governed by such equations.