Analytic Solution for A ==> B

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A ----> B

The differential equation to be solved is

The technique will be separation of variables.

The solution is easily accomplished by direct integration.

Some parameters will be defined so that MathCad can plot the solution. I will only use 20 time points and each time point will be 1/40 of a second.

This is the integrated equation

Next I will perform some minor algebra.

This is a form of the equation that can be graphed.

Having obtained A, getting B is trivial by subtraction, however, I will derive it explicitly.

This is the new equation in B.

Since we know the answer to A, we will substitute it in.

but

This line shows the substitution.

This line is separation of variables.

And integration

This is the integrated result and is graphed below.