Time derivative for time dependent production and decay of protein (This is taken from An Introduction to Systems Biology. Alon. 2007.)
A gene Y with simple regulation has a time-dependent production rate B(t) and a time-dependent degradation rate a(t). Solve for its concentration as a function of time. apparently, the solution must be verified by taking the time derivative. Please can someone help me with this exercise?
The ODE for a gene with simple non-time dependent production and decay looks like this:
dY/dt = B - aY
Where Y is the protein being produced by gene Y.
(thanks for reading my question!) 
 A: Here is an outline for solving your ODE $Y'(t)=B(t)-a(t)Y(t)$:


*

*First solve the simpler ODE $f'(t)=-a(t)f(t)$. This is a separable equation and easy to solve using standard methods. (I assume you have access to some material on ODEs.) It is often a good idea to solve the corresponding homogeneous equation first.

*Look for a solution to your original ODE in the form $Y(t)=u(t)f(t)$. If $f$ solves $f'=-af$ and $Y$ solves $Y'=B-aY$, you get an ODE for $u$. You don't need to use the solution from step 1, only the ODEs.

*Solve this new equation for $u$. It is again a simple ODE.

*You now know $f$ and $u$, so you can write your full solution $Y(t)=u(t)f(t)$.

*There will be an unknown constant in the solution. Solve the constant if you know $Y(0)$ (or something similar).

*Take the derivative of $Y$ and check that it satisfies your original ODE.


If you want, you can compare your solution to the one given by a computer algebra system. It may look different even if it is the same. For your ODE, see this.
