When do you use differential equations? I have a problem whose answer I find rather questionable in my textbook.

A psychologist has determined that the initial probability of success for a given exam is 0.02, and that the maximum amount learned is 0.86. 
After studying 1 month, probability of success increases to 0.23
Within how many months studying will the probability reach 0.70?

The answer to this exercise is formulated as a differential function.
I do not see the need to differentiate as there is no indication of non-linearity.
Can't I simply go
$$\frac{0.23 - 0.2}{1\text{ (month)}}$$
Rate of learning is $0.21/\text{month}$, so it will take
$$\frac{0.70-0.2}{0.21} = 3.24$$
months to reach 0.70.
How is this process incorrect?
 A: If that's all it says (with no previous discussion of models for learning), this is a very poorly formulated exercise.  There is no
indication that the probability of success should satisfy any particular kind of differential equation.  
You could say that this is 
also a feature of real-life research: nature does not tell you what kind of 
model to use.  However, in a real situation a researcher might look at a bunch of
data, try to find a reasonable model, and test that model against more data.  Here you  don't have that.
A: There is no mention of "non-linearity" but there is also no mention of "linearity"!  NOT given in the statement of the problem but generally true is that learning obeys a "logistic equation".  That is a differential equation of the form $dx/dt= Ax(T- x)$.  That is, when $x$ is close to $0$, when you are just starting to learn a field, the rate of learning is low, as you become more familiar with the subject, learning becomes easier, but eventually, as $x$ nears $T$, the total information of the field, learning becomes harder again.   
