# Finding an expression for $\dfrac{dx}{dt}$ by solving the initial value problem

The effectiveness of a police force may be measured by its clearance rate: the number of charges laid in a month divided by the total number of unsolved crimes.

In Arachnid Boy's home town, new crimes are reported roughly $$20$$ times per month, and while Arachnid boy is in town, the police clearance rate is $$40\%$$. Arachnid Boy comes back from his holiday and finds there are $$100$$ unsolved crimes.

Let $$x$$ be the number of unsolved crimes at the start of month $$t$$ , with $$t=0$$ representing the first month that Arachnid Boy is back from his holiday. What is the value of $$\dfrac{dx}{dt}$$?

All I've gotten to is

$$\dfrac{dx}{dt} = 100 - 8x$$

although I know that this expression is incorrect. I've tried to consider a linear relationship and solving for constants using the initial conditions, however I highly doubt this is the correct way to attempt this question.

Any help or guidance is greatly appreciated!

Each month, $$20$$ new cases come in. That means $$\frac{dx}{dt}$$ gets a $$+20$$ contribution from that. Also, each month, they clear $$40\%$$ of all cases, so $$\frac{dx}{dt}$$ gets a $$-0.4x$$ contribution from that. These are the only pieces of information that we are given regarding how the number of cases changes from month to month.
Which is to say, $$\frac{dx}{dt}=20-0.4x$$ The $$100$$ cases is an initial value, and doesn't affect this expression for $$\frac{dx}{dt}$$ at all. Of course, if you want to solve this initial value problem, the $$100$$ must be used.