Originally Posted by

**mathman89**
So if you have some functions A_n dependent on only one variable t and there is a matrix B_nk: dA_n/dt = B_nk A_k (where there is a summation in index k) if B_nk is a constant matrix then knowing its eigenvalues will give

an easy solution to the system. Say B has eigenvectors x,y and eigenvalues ex, ey. Take a function u(t) and multiply by x to get something two-dimensional. Bx*u(t)=ex*x*u(t) so plugging it into the differential equation will give x*u'(t)=x*ex*u(t) which clearly has a solution u=exp(ex*t)*K1. Since the system is linear sums of solutions are solutions. If the eigenvalues are different the solutions exp(t ey)y*Ky and exp(t ex)x*Kx are linearly independent and so the system has a complete solution.

The eigenvalues here have units inverse time and if they are negative i guess they could be interpreted as a relaxation time.

If the t0, t1 and stuffs are just constants then this applies right away. If they are functions of t for instance then you'd have to linearize at some point in order to use this method.