Trouble solving an ordinary differential equation

  • Thread starter Hypatio
  • Start date
  • #1
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Homework Statement



Find the appropriate equation.

Homework Equations



So there we have the ordrinary differential equation

[itex]\frac{d M}{dt}=k_1M-k_2(1-M)=A\exp \left (-\frac{E}{T} \right )M-B\exp \left ( -\frac{F}{T} \right )(1-M)[/itex]

The goal is to solve the differential equation. It turns out the solution should be something like this:

[itex]M=\frac{k_2}{K_1+k_2}+\frac{k_1}{K_1+k_2}\exp -(k_1+k_2)t[/itex]

although I think there may be a typo around the last exp (im not sure if t is inside the exponent or not)


The Attempt at a Solution



After integrating over t I get

[itex]M=Mt(k_1+k_2)-k_2 t[/itex]

But I'm not even sure this is the correct integral of the equation as I don't know how the supposed solution follows from this.
 

Answers and Replies

  • #2
lanedance
Homework Helper
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You integration over t is not correct, you need to remember M=M(t) is a function of t.

To integrate directly the DE must be separable, this one is not but i think it can be made so with a simple substitution
 
  • #3
lanedance
Homework Helper
3,304
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An example of a separable DE is as follows
[tex]
\frac{dx}{dt} = kx
[/tex]

rearranging and integrating gives
[tex]
\int\frac{dx}{x} = k\int dt
[/tex]
[tex]
ln(x) = kt+c
[/tex]
[tex]
x = e^{c}e^{kt}
[/tex]
 
  • #5
lanedance
Homework Helper
3,304
2
or write your ODE as
M' = aM+b

and make the subsitution
N = aM+b
 

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