ODE Linear nth Order Numeric (Not in Base Package)

Solves an nth order homogeneous linear differential equation with constant coefficients in numeric form. Details

A is the vector of coefficients of the different derivatives of a function x(t), starting with the coefficient of the lowest order term. The coefficient of the highest order derivative is assumed to be equal to 1.0 and does not need to be entered.
X0 is the vector of the start condition . There is a one-to-one relation between the components of X0 and X.
number of points is the number of equidistant time points between time start and time end. The default is 10.
time start is the start point of the ODE. The default is 0.
time end is the end point of the time interval under investigation. The default is 1.0.
Times is an array representing the time steps. The method yields equidistant time steps between time start end time end.
X is the vector of the solution x at the equidistant time points as specified in the Times array.
errors are produced by using the wrong inputs X, X0, and F(X,t).

ODE Linear nth Order Numeric Details

Consider the nth order linear homogeneous differential equation

where 0 represents the more general value of time start. There is a strong connection between the equation

and the zero finding problem

The n zeroes of the last equation determine the structure of the solution of the ODE. If we have n distinct complex zeroes , the general solution of the nth order differential equation can be expressed by

The unknowns can be determined by the start condition

Note  The case of repeated eigenvalues is more complex and is not treated here. An error code of –23017 is given if this happens.

By convention, the value of the highest coefficient is taken as 1.0, and does not need to be entered in the A control. The other coefficients are entered starting with the lowest order coefficient.

To solve the differential equation

x'' – 3 x' + 2 x = 0

with the I.C. as with x(0) = 2 and x'(0) = 3

enter

A = [2, –3] and X0 = [2, 3]