Homogeneous Equations With Constant Coefficients
The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form, for possibly-complex values of . The exponential function is one of the few functions that keep its shape after differentiation. In order for the sum of multiple derivatives of a function to sum up to zero, the derivatives must cancel each other out and the only way for them to do so is for the derivatives to have the same form as the initial function. Thus, to solve
we set, leading to
Division by e zx gives the nth-order polynomial
This algebraic equation F(z) = 0, is the characteristic equation considered later by Gaspard Monge and Augustin-Louis Cauchy.
Formally, the terms
of the original differential equation are replaced by zk. Solving the polynomial gives n values of z, z1, ..., zn. Substitution of any of those values for z into e zx gives a solution e zix. Since homogeneous linear differential equations obey the superposition principle, any linear combination of these functions also satisfies the differential equation.
When these roots are all distinct, we have n distinct solutions to the differential equation. It can be shown that these are linearly independent, by applying the Vandermonde determinant, and together they form a basis of the space of all solutions of the differential equation.
Examples |
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has the characteristic equation This has zeroes, i, −i, and 1 (multiplicity 2). The solution basis is then This corresponds to the real-valued solution basis |
The preceding gave a solution for the case when all zeros are distinct, that is, each has multiplicity 1. For the general case, if z is a (possibly complex) zero (or root) of F(z) having multiplicity m, then, for, is a solution of the ODE. Applying this to all roots gives a collection of n distinct and linearly independent functions, where n is the degree of F(z). As before, these functions make up a basis of the solution space.
If the coefficients Ai of the differential equation are real, then real-valued solutions are generally preferable. Since non-real roots z then come in conjugate pairs, so do their corresponding basis functions xkezx, and the desired result is obtained by replacing each pair with their real-valued linear combinations Re(y) and Im(y), where y is one of the pair.
A case that involves complex roots can be solved with the aid of Euler's formula.
Read more about this topic: Linear Differential Equation
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