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.)

And we can integrate both sides of the equation using the following substitution:

And replacing we got:

If we multiply both sides by -1 we got:

And using exponential in both sides of the equation we got:

And solving for P we got:

And replacing we got:

We can use the condition and we got:

And we see that and replacing we got:

Step-by-step explanation:

For this case we aasume the following differential equation:

Is a separable differential equation so we can do the following procedure:

And we can integrate both sides of the equation using the following substitution:

And replacing we got:

If we multiply both sides by -1 we got:

And using exponential in both sides of the equation we got:

And solving for P we got:

And replacing we got:

We can use the condition and we got:

And we see that and replacing we got:

power property.