A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders.
is constant and has n linearly independent eigenvectors, this differential equation has the following general solution, where λ1, λ2, …, λn are the eigenvalues of A; u1, u2, …, un are the respective eigenvectors of A; and c1, c2, …, cn are constants.
then the Magnus expansion reduces to leading order, and the general solution to the differential equation is where
By use of the Cayley–Hamilton theorem and Vandermonde-type matrices, this formal matrix exponential solution may be reduced to a simple form.
[1] Below, this solution is displayed in terms of Putzer's algorithm.
[2] The matrix equation with n×1 parameter constant vector b is stable if and only if all eigenvalues of the constant matrix A have a negative real part.
The steady state x* to which it converges if stable is found by setting thus yielding assuming A is invertible.
Thus, the original equation can be written in the homogeneous form in terms of deviations from the steady state, An equivalent way of expressing this is that x* is a particular solution to the inhomogeneous equation, while all solutions are in the form with
In the n = 2 case (with two state variables), the stability conditions that the two eigenvalues of the transition matrix A each have a negative real part are equivalent to the conditions that the trace of A be negative and its determinant be positive.
has the matrix exponential form evaluated using any of a multitude of techniques.
Note the algorithm does not require that the matrix A be diagonalizable and bypasses complexities of the Jordan canonical forms normally utilized.
Higher order matrix ODE's may possess a much more complicated form.
The process of solving the above equations and finding the required functions of this particular order and form consists of 3 main steps.
Brief descriptions of each of these steps are listed below: The final, third, step in solving these sorts of ordinary differential equations is usually done by means of plugging in the values calculated in the two previous steps into a specialized general form equation, mentioned later in this article.
To solve a matrix ODE according to the three steps detailed above, using simple matrices in the process, let us find, say, a function x and a function y both in terms of the single independent variable t, in the following homogeneous linear differential equation of the first order, To solve this particular ordinary differential equation system, at some point in the solution process, we shall need a set of two initial values (corresponding to the two state variables at the starting point).
The first step, already mentioned above, is finding the eigenvalues of A in The derivative notation x′ etc.
Once the coefficients of the two variables have been written in the matrix form A displayed above, one may evaluate the eigenvalues.
, multiplied by some constant λ, is subtracted from the above coefficient matrix to yield the characteristic polynomial of it, and solve for its zeroes.
Applying further simplification and basic rules of matrix addition yields Applying the rules of finding the determinant of a single 2×2 matrix, yields the following elementary quadratic equation, which may be reduced further to get a simpler version of the above, Now finding the two roots,
of the given quadratic equation by applying the factorization method yields The values
In some cases, say other matrix ODE's, the eigenvalues may be complex, in which case the following step of the solving process, as well as the final form and the solution, may dramatically change.
As mentioned above, this step involves finding the eigenvectors of A from the information originally provided.
, we have Simplifying the above expression by applying basic matrix multiplication rules yields All of these calculations have been done only to obtain the last expression, which in our case is α = 2β.
Now taking some arbitrary value, presumably, a small insignificant value, which is much easier to work with, for either α or β (in most cases, it does not really matter), we substitute it into α = 2β.
Doing so produces a simple vector, which is the required eigenvector for this particular eigenvalue.
In our case, we pick α = 2, which, in turn determines that β = 1 and, using the standard vector notation, our vector looks like Performing the same operation using the second eigenvalue we calculated, which is
The process of working out this vector is not shown, but the final result is This final step finds the required functions that are 'hidden' behind the derivatives given to us originally.
The equation which involves all the pieces of information that we have previously found, has the following form: Substituting the values of eigenvalues and eigenvectors yields Applying further simplification, Simplifying further and writing the equations for functions x and y separately, The above equations are, in fact, the general functions sought, but they are in their general form (with unspecified values of A and B), whilst we want to actually find their exact forms and solutions.
, which plays the role of starting point for our ordinary differential equation; application of these conditions specifies the constants, A and B.
The above problem could have been solved with a direct application of the matrix exponential.