Trouble solving an ordinary differential equation

[Hypatio]

Homework Statement

Find the appropriate equation.

Homework Equations

So there we have the ordrinary differential equation

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

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

$M=\frac{k_2}{K_1+k_2}+\frac{k_1}{K_1+k_2}\exp -(k_1+k_2)t$

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

$M=Mt(k_1+k_2)-k_2 t$

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.

 

[lanedance]

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

 

[lanedance]

An example of a separable DE is as follows
$$\frac{dx}{dt} = kx$$

rearranging and integrating gives
$$\int\frac{dx}{x} = k\int dt$$
$$ln(x) = kt+c$$
$$x = e^{c}e^{kt}$$

 

[bigfooted]

Exactly.
The standard method for this nonhomogeneous first order ode would be to use an integrating factor.
see http://en.wikipedia.org/wiki/Integrating_factor
and other places.
write your ode as: M' + aM = b

 

[lanedance]

or write your ODE as
M' = aM+b

and make the subsitution
N = aM+b