We begin by referring to our equations (1) and (2) as shown here. We re-arrange the equations so that all terms in

*C*are on the left side of (1) and all terms in

_{A}*C*are on the left side of (2). In doing so, we treat d/d

_{R}*t*as if it were just another factor, which may seem odd, but it's accurate.

The only variables in these two equations are

*C*,

_{A}*C*, and time,

_{R}*t*. All other parameters are constants that we have already calculated. We must eliminate

*C*from the (3) so as to obtain an equation in

_{R}*C*and

_{A}*t*alone, which we can then solve. We eliminate

*C*by multiplying (3) by the same factor that we observe on the left side of (4).

_{R}We do the same thing for

*C*, arriving at a differential equation in only

_{R}*C*and

_{R}*t*.

At this point we pause to check the equations by considering how they behave as time approaches infinity, as shown here, and we find that they appear to behave correctly. Both equations have solutions of the same form: a constant plus two decay terms.

We insert values for

*m*, γ,

_{p}^{C14}*M*,

_{A}*M*, and

_{R}*m*and obtain values for all five constants in our solution. The coefficients α and β dictate how rapidly the concentration evolves with time. We have α = 0.0574 and β = 0.000122. We see that α is close to the fraction of the atmospheric carbon that is exchanged with the reservoir every year, while β is close to the decay rate of carbon-14. We have 1/α = 17 yr, which is the time constant of exchanges between the atmosphere and the reservoir, and 1/β = 8,200 yrs, which is the time constant of accumulation of carbon-14 in the reservoir. The weighting factors

_{e}*k*and

_{1}*k*are −0.2 and −0.8 respectively. Together, they add up to −1.0 ppt. The constant term is the equilibrium value of

_{2}*C*, which comes out as 1.0 ppt, which is what we expect, because we chose the value of

_{A}*m*and

_{e}*M*to make sure that the equilibrium concentration would be 1.0 ppt. Our equation for

_{R}*C*is as follows, where concentration is in ppt and time is in years.

_{A}*C*= 1.0 − 0.2

_{A}*e*

^{−t/17}− 0.8

*e*

^{−t/8200}.

At

*t*= 0, we have

*C*= 0 ppt, and when

_{A}*t*→∞,

*C*→1.0 ppt, as we expect. The 17-yr decay term represents the flow of carbon-14 into the reservoir. The 8200-yr term represents the accumulation of carbon-14 in the reservoir. We obtain a similar solution for

_{A}*C*.

_{R}The coefficients α' and β' are the same as α and β. But the weighting factors are different, as is the constant term. Our equation for

*C*is,

_{R}*C*= 0.8 + 0.002

_{R}*e*

^{−t/17}− 0.802

*e*

^{−t/8200}.

At

*t*= 0, we have

*C*= 0 ppt, and d

_{R}*C*/d

_{R}*t*=

*k*α+

_{1}*k*β = 0.0 ppt/yr, and d

_{2}^{2}

*C*/d

_{R}*t*

^{2}=

*k*α

_{1}^{2}+

*k*β

_{2}^{2}> 0, all of which we expect, and also when

*t*→∞, we have

*C*→0.8 ppt, which is one of our starting assumptions.

_{R}We are pleased to have an analytic solution for

*C*and

_{A}*C*. A numerical solution to the differential equations turns out to be unstable for time steps greater than ten years. We want to plot

_{R}*C*and

_{A}*C*over a hundred thousand years. Ten thousand steps are cumbersome in a spreadsheet. Furthermore, the analytic solution us gives more insight into the way the parameters of the carbon cycle interact to govern its behavior.

_{R}
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