This type of differential equation is a third-order linear homogeneous equation. It can be solved using techniques such as the characteristic equation method. By assuming a solution of the form , where is a constant, you would substitute this into the equation to obtain the characteristic polynomial . Solving for provides the roots that help construct the general solution.

To avoid common missteps, remember to double-check your characteristic polynomial setup. It’s easy to make a small algebraic mistake when substituting derivatives or coefficients. Additionally, consider the nature of the roots: distinct, repeated, or complex—as they will dictate the form of your general solution. If you have complex roots, be sure to use Euler’s formula to express your solutions in a manageable format!