Limits of Differential Equations

In summary, the equation for x in part a of the homework statement is based off of a graph, but is incorrect.
  • #1
2
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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. [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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  • #2
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. [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?
 
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  • #3
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. [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
Never mind I had it right early. Thanks anyway everyone!
 

What are differential equations?

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model many physical, biological, and economic systems.

What are the limits of differential equations?

The limits of differential equations are the situations in which they may not accurately model a system. These can include non-linear systems, chaotic systems, and systems with changing parameters.

How are differential equations used in science?

Differential equations are used in science to model and understand complex systems. They are used in physics, chemistry, biology, economics, and many other fields to make predictions and analyze data.

What are some common techniques for solving differential equations?

Some common techniques for solving differential equations include separation of variables, substitution, and using power series. Numerical methods, such as Euler's method and Runge-Kutta methods, are also frequently used.

What are some real-world applications of differential equations?

Differential equations have many real-world applications, including predicting weather patterns, modeling population growth, understanding the spread of diseases, and designing efficient structures in engineering. They are also used in financial modeling and in developing control systems for robots and other machines.

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