Limits of Differential Equations

  • #1
2
0

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. [Broken] 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. [Broken] 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.
 
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Answers and Replies

  • #2
34,671
6,382

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. [Broken] 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. [Broken] 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:
  • #3
ehild
Homework Helper
15,543
1,909

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. [Broken] 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. [Broken]
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.
 
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  • #4
2
0
Never mind I had it right early. Thanks anyway everyone!
 

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