Question

Solve dX/dt = A-BX,X(0)=0,where A and B are positive constants. Find X when t->infinity and t when the function reaches half of the limit's value.

Answer 1

Step-by-step explanation:

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Answer 2

Given : dX/dt  = A - BX

X(0) = 0  A and B are positive constant

To Find : X   when t → ∞

t when the function reaches half of the limit's value.

Solution:

dX/dt  = A - BX

=> dX/(A - BX)  = dt

Integrating both sides

=> (-1/B) ln ( A - BX)  = t  + C  

=>  ln ( A - BX)  = -Bt  + C

X(0) = 0

=> ln (A) = C

=>  ln ( A - BX)  = -Bt  + ln A

=>  ln ( A - BX)  - ln A  = - Bt

=> ln (A - BX)/A  = - Bt

A - BX  = A $e^{-Bt}$

t → ∞   $e^{-Bt}$ = 0

=> A - BX  = 0

=> X  = A/B

when the function reaches half of the limit's value.

=> X = A/2B

A - B(A/2B)  = A $e^{-Bt}$

=> A/2 = A$e^{-Bt}$

=> 1/2  = $e^{-Bt}$

=> ln 2 = Bt

=> t = ln 2/ B

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