Mathematics From Futurama We at D.O.O.P are trying to mathematically model a rocket ship fueled by your employee Leela's pet Nibbler's pooped Black matter. Obviously this rocket ship is fueled by black matter which along with the ship's combustion chamber has some special properties.
Black matter Properties
We have discovered that there are two kinds of black matter the one naturally made (by Nibbler's poop) and the other which is  synthetically made in our special rocket ship combustion engine. The Synthetically made Black matter exponentially decays at a rate of $r$, until all its mass eventually becomes nothing and is discarded. 
Rocket Ship Properties
When any Black matter is used as fuel in the combustion chamber. It produces black matter (synthetic kind) according to following equation.
$$\dfrac {dP_a} {dt}\leq \dfrac {dP_c} {dt}=\left( 1-\dfrac {\alpha } {100}\right)B$$
Here $B$ represents the amount of Black matter currently in the combustion chamber. $\alpha$ is a percentage, a controlling mechanism in the ship to control production. $P_c$ is an upper bound of the new Black matter production (capacity), but we discover there is an inefficiency in the system such that the actual rate of Black matter production is in fact $P_{a}$. 
Since we want to travel as fast as we can, as soon as any new Black matter $P_{a}$ is produced we add that to $B$ and we are able to do all of this in infinitesimally small amount of time(continuously).
We invite you to scientifically examine and model the processes of this rocket ship along with say $B_0$ amount of initial natural black matter. How can we model or represent this system with the least amount of equations while capturing the essence of the whole problem ?
From the desk of Zapp Brannigan
"And like all my plans, it's so simple an idiot could have devised it!"
     Edit:

Solution attempt
Assuming $B_0$ to be the initial amount of black matter available. We start undertaking combustion with this initial amount $B_0$ we produce more black matter at the rate of $\dfrac {dP_a} {dt}\leq \dfrac {dP_c} {dt}=\left( 1-\dfrac {\alpha } {100}\right)B_{0}$. As $dt$ time period passes by we take the new $P_a$ amount produced and add it to $B_{0}$. We also observe that this newly created synthetic black matter $P_a$ is exponentially decaying. I am having trouble figuring out how to put these relations together so both of these processes can be carried out simultaneously.I'd be happy with if you wish, only consider the case when $P_a$ and $P_c$ are the same.
 A: I think you have told us that $B$ is changing with respect to time for two reasons. One has positive influence: new dark matter is produced by your engine. The other has negative influence: the dark matter decays exponentially. I assume that it is this decay that powers the engine, because I don't see anything else explaining how the fuel is burned for the purposes of propulsion.
So
$$\begin{align}
\frac{dB}{dt} & = \frac{dP_a}{dt} - rB\\
\end{align}
$$
Assuming $\frac{dP_a}{dt}=\frac{dP_c}{dt}$,
$$\begin{align}
\frac{dB}{dt} & =\left(1-\frac{\alpha}{100}\right)B - rB\\
&=\left(1-\frac{\alpha}{100}-r\right)B
\end{align}
$$
This equation has solution $B=B_0e^{\left(1-\frac{\alpha}{100}-r\right)t}$
With $\frac{dP_a}{dt}<\frac{dP_c}{dt}$, of course much more is possible. In the extreme, $\frac{dP_a}{dt}=0$, and $B=B_0e^{-rt}$.
So $$B_0e^{-rt}\leq B(t)\leq B_0e^{\left(1-\frac{\alpha}{100}-r\right)t}$$ Any curve that fits between these two and never has a steeper relative growth rate than the upper bound nor a less steep relative growth rate than the lower bound is a possible solution, depending on how exactly $\frac{dP_a}{dt}$ differs from $\frac{dP_c}{dt}$.
