Limits of Differential Equations

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Homework Help Overview

The discussion revolves around finding the limit of a differential equation given by (dx/dt) = k(a-x)(b-x) with the initial condition x(0)=0. The participants are exploring two specific cases: one where 0

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss separating the equation and solving for x, noting challenges with the resulting equations. There are questions about the correctness of the derived expressions and the implications of the initial conditions on the limits.

Discussion Status

Some participants have provided feedback on the attempts made, questioning the accuracy of certain steps and suggesting re-evaluation of the derived equations. There is an ongoing exploration of the implications of the initial conditions and the limits as t approaches infinity, but no consensus has been reached.

Contextual Notes

Participants are working under the constraints of the problem statement, with no additional equations provided. There is a focus on the limits of the differential equation without a clear resolution or final answer being presented.

olive.p
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Homework Statement


I need help finding the limit of the differential equation.
(dx/dt) = k(a-x)(b-x) that satisfies x(0)=0
assuming
a) 0<a<b and find the limit as t->infinity of X(t)
b) 0<a=b and find the limit as t->infinity of X(t)

Homework Equations


none

The Attempt at a Solution



I separated the equation in part a and attempted to solve for x and got a nasty equation
http://www4b.wolframalpha.com/Calculate/MSP/MSP115222ac5cd5fd1ghhf0000016hfg120ch979ad9?MSPStoreType=image/gif&s=20&w=156.&h=41. then I solved for c and found it to be c=-(a/b). I plugged that in for c and got:
http://www4f.wolframalpha.com/Calculate/MSP/MSP49220eh2769a9a2d53700001g9fiib9hd1eh2c3?MSPStoreType=image/gif&s=49&w=159.&h=50. I don't know how to take it further.
I believe that the answer to part a is a based of a graph, but I am unable to prove it.
Thanks in advance.
 
Last edited by a moderator:
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olive.p said:

Homework Statement


I need help finding the limit of the differential equation.
(dx/dt) = k(a-x)(b-x) that satisfies x(0)=0
assuming
a) 0<a<b and find the limit as t->infinity of X(t)
b) 0<a=b and find the limit as t->infinity of X(t)

Homework Equations


none

The Attempt at a Solution



I separated the equation in part a and attempted to solve for x and got a nasty equation
http://www4b.wolframalpha.com/Calculate/MSP/MSP115222ac5cd5fd1ghhf0000016hfg120ch979ad9?MSPStoreType=image/gif&s=20&w=156.&h=41. then I solved for c and found it to be c=-(a/b). I plugged that in for c and got:
http://www4f.wolframalpha.com/Calculate/MSP/MSP49220eh2769a9a2d53700001g9fiib9hd1eh2c3?MSPStoreType=image/gif&s=49&w=159.&h=50. I don't know how to take it further.
I believe that the answer to part a is a based of a graph, but I am unable to prove it.
Thanks in advance.
Did you check the solution you got for x in your first equation above?
 
Last edited by a moderator:
olive.p said:

Homework Statement



The Attempt at a Solution



I separated the equation in part a and attempted to solve for x and got a nasty equation
http://www4b.wolframalpha.com/Calculate/MSP/MSP115222ac5cd5fd1ghhf0000016hfg120ch979ad9?MSPStoreType=image/gif&s=20&w=156.&h=41. then I solved for c and found it to be c=-(a/b). I plugged that in for c and got:
http://www4f.wolframalpha.com/Calculate/MSP/MSP49220eh2769a9a2d53700001g9fiib9hd1eh2c3?MSPStoreType=image/gif&s=49&w=159.&h=50.

The last equation is wrong. Why did you change the second exponent?

You can replace ## e^{akt} e^{-bkt } = e^{(a-b)kt } ## in the first equation. The value c=-a/b is right. Just plug in for c.
 
Last edited by a moderator:
Never mind I had it right early. Thanks anyway everyone!
 

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