Question

Given: 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.)

Answer 1

Answer:

$P=M(1-e^{-kt})$

Step-by-step explanation:

The relation between the variables is given by

$\frac{dP}{dt} = k(M- P)$

This is a separable differential equation. Rearranging terms:

$\frac{dP}{(M- P)} = kdt$

Multiplying by -1

$\frac{dP}{(P- M)} = -kdt$

Integrating

$ln(P-M)=-kt+D$

Where D is a constant. Applying expoentials

$P-M=e^{-kt+D}=Ce^{-kt}$

Where $C=e^{D}$, another constant

Solving for P

$P=M+Ce^{-kt}$

With the initial condition P=0 when t=0

$0=M+Ce^{-k(0)}$

We get C=-M. The final expression for P is

$P=M-Me^{-kt}$

$P=M(1-e^{-kt})$

Keywords: performance , learning , skill , training , differential equation