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This paper is concerned with the linear ODE in the form$y'(t)=\lambda\rho(t)y(t)+b(t)$, $\lambda <0$ which represents a simplifiedstorage model of the carbon in the soil. In the first part, we show that, for aperiodic function $\rho(t)$, a linear drift in the coefficient $b(t)$ involvesa linear drift for the solution of this ODE. In the second part, we extend theprevious results to a classical heat non-homogeneous equation. The connectionwith an analytic semi-group associated to the ODE equation is considered in thethird part. Numerical examples are given.

Large time behavior of differential equations with drifted periodic coefficients modeling Carbon storage in soil