Analytic Solution for A <=> B

     k1
A <==> B
     k2

This is the reversible mechanism for species A converting to B and vice versa

There are two differential equations.

Here we will define constants and initial conditions.

Using a matrix solution entirely and let MathCad do the work.

This is the matrix of coefficients

This is the root of the characteristic equation (also called auxiliary equation).

These are the eigenvectors and they are not unique.

This is an alternative set, harder to find, easier to use.

The general solution is:

Solve for p1 and p2 from initial conditions,

<= General solution

While the solution to the unknown constants looks very complex, it is just algebra. The initial condition is time zero, so e to the power of zero is 1.

Final symbolic solution