Limits of Differential Equations

[olive.p]

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.

 

[Mark44]

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?

 

[ehild]

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.

 

[olive.p]

Never mind I had it right early. Thanks anyway everyone!