Acpw help please!!! (1 Viewer)

[YJ90]

I need help with these 2 questions please, thanks in advance:

1) dP/dt = kP(1 - RP) where k, R are constants.

Show by differentiation that P = I/[RI + (1 - RI)e^-kt] , where I is the intial population (a constant).

2) dP/dt = 0.001P(100 - P)

Show by substitution that P = 100/[1+(1/k).e^-0.1t]

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[Trebla]

I need help with these 2 questions please, thanks in advance:

1) dP/dt = kP(1 - RP) where k, R are constants.

Show by differentiation that P = I/[RI + (1 - RI)e^-kt] , where I is the intial population (a constant).

2) dP/dt = 0.001P(100 - P)

Show by substitution that P = 100/[1+(1/k).e^-0.1t]

1)
P = I.[RI + (1 - RI)e^-kt]^-1
=> [RI + (1 - RI)e^-kt] = I/P
=> (1 - RI)e^-kt = I/P - RI
dP/dt = I.(1 - RI)(-ke^-kt)(-1)[RI + (1 - RI)e^-kt]^-2
= kI(1 - RI)(e^-kt)P²/I²
= k(I/P - RI)P²/I
= k(P - RP²)
= kP(1 - RP)

2)
If P = 100/[1+(1/k).e^-0.1t]
=> [1+(1/k).e^-0.1t] = 100/P
=> (1/k).e^-0.1t = 100/P - 1
dP/dt = 100(1/k)(-0.1)e^-0.1t(-1)[1+(1/k).e^-0.1t]^-2
= 10(1/k)e^-0.1t/[1+(1/k).e^-0.1t]²
= 10(1/k)e^-0.1tP²/10 000
= (1/k)e^-0.1tP²/1000
= (100/P - 1)P²/1000
= (100P - P²)/1000
= 0.001P(100 - P)

 

[YJ90]

Thanks for the help "Trebla"