Anonymous
Anonymous asked in Science & MathematicsMathematics · 4 weeks ago

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.)

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  • ?
    Lv 7
    4 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) )

    Attachment image
  • 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))

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