In a previous exercise we formulated a model for learning in the form of the differential equation dp dt = k(m − p) where p(t) measures the performance of someone learning a skill after a training time t, m is the maximum level of performance, and k is a positive constant. solve this differential equation to find an expression for p(t). (use p for p(t). assume that p(0) = 0.) incorrect: your answer is incorrect.

I assume you mean

   

ANSWER
An expression for P(t) is

   

EXPLANATION
This is a separable differential equation. Treat M and k as constants. Then we can divide both sides by M - P to get the P term with the differential dP and multiply both sides by dt to separate dt from the P terms

   

Integrate both sides of the equation.

   

Note that for the left-hand side, u-substitution gives us 

   

hence why 

Now we use the definition of the logarithm to convert into exponential form.

The definition is 

   

so applying it here, we get

   

Exponent properties can be used to address the constant C. We use  here:

   

If we assume that P(0) = 0, then set t = 0 and P = 0

   

Substituting into our original equation, we get our final answer of

   

Ithink it’s b, hope this .