What is the solution of characteristic root equation?

Eigenroot method is a general method for solving linear differential equations with constant coefficients in mathematics.

The characteristic root method can also find the general term formula through the recursive formula of sequence (that is, the difference equation must be linear), and its essence is the same as that of differential equation. For example, it is called second-order homogeneous linear difference equation: weighted characteristic equation.

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The equations constructed by the two methods are only different in the coefficient of the first term, and they are opposite. So the solutions of the two equations should be opposite. From this point of view, although there are additions and subtractions in form, the series constructed by the two methods are actually exactly the same. Obviously, the process of finding the characteristic equation in the characteristic root method is more direct and easier to remember.

Although it is not clear why this equation is constructed so casually, the idea of characteristic root method undoubtedly makes the process more concise on the premise that both solutions are established.

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