# solve the differential equation ?

We have 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.)

### 2 Answers

- ?Lv 74 weeks ago
dP/dt = k(M − P)

dP/dt = -k(P - M)

1/(P - M) dP = -k dt

ln|P - M| = -kt + lnC

No big deal lnC is just a is an arbitrary constant of integration

ln| P - M | = -kt + lnC

ln| ( P - M )/lnC | = -kt

P - M = Ce^(-kt)

P = M + Ce^(-kt)

Initial value conditions

0 = M + Ce^(-k*0)

C = -M

P = M - Me^(-kt)

P(t) = M( 1 - e^(-kt) )

- 4 weeks ago
dP/dt = k * (M - P)

dP / (M - P) = k * dt

dP / (P - M) = -k * dt

Integrate

ln|P - M| = -kt + C

P - M = e^(-kt + C)

P = M + e^(-kt) * e^C

P = M + A * e^(-kt)

P(t) = M + A * e^(-kt)

P(0) = 0

0 = M + A * e^(-k * 0)

0 = M + A * e^(0)

0 = M + A * 1

0 = M + A

A = -M

P(t) = M - M * e^(-kt)

P(t) = M * (1 - e^(-kt))