Thread: ODE solution of exponential form

1. ODE solution of exponential form

Hello everybody

I'm trying to solve the following differential equation derived from a first order kinetic process:

$\displaystyle \tau {dm(t) \over dt} = m(\infty) - m(t)$

where $\displaystyle m(\infty)$ is the steady-state of $\displaystyle m(t)$, that I guess, for the moment could be treated as a constant.

I know that the solution to the ODE is:

$\displaystyle m(t) = m(\infty) -\big(m(\infty)-m(0)\big ) \exp({-t/\tau} )$

being $\displaystyle m(0)$ the initial condition.

I've tried to solve the equation with a piece of paper, but get stuck in the process. Im principle, this equation should be easily solved by separation of variables...

What I tried so far:
First, separation of variables

$\displaystyle \frac{dm(t)}{m(\infty) - m(t)} = \frac{dt}{\tau}$

applying the integration limits:

$\displaystyle \int\limits_{0}^{t} \frac{dm(t)}{m(\infty) - m(t)} =\int\limits_{0}^{t} \frac{dt}{\tau}$

solving the integral

$\displaystyle \log\frac{m(\infty) - m(t)}{m(\infty) - m(0)} =\frac{t}{\tau}$

and applying exponential to both sides of the equation leads to:

$\displaystyle \frac{m(\infty) - m(t)}{m(\infty) - m(0)} =\exp(t/\tau)$

organizing...
$\displaystyle m(\infty) - m(t) = \big(m(\infty) - m(0)\big) \exp(t/\tau)$

finally, solving for $\displaystyle m(t)$

$\displaystyle m(t) = m(\infty) - \big(m(\infty) - m(0)\big) \exp(t/\tau)$

which is almost the solution, but the exponential is raised to $\displaystyle t/\tau$ and not to $\displaystyle -t/\tau$ as it should be.

I would appreciate if somebody would point out where is the mistake in the calculation

Thanks

Jose

2. Re: ODE solution of exponential form

Originally Posted by jguzman
Hello everybody

I'm trying to solve the following differential equation derived from a first order kinetic process:

$\displaystyle \tau {dm(t) \over dt} = m(\infty) - m(t)$

where $\displaystyle m(\infty)$ is the steady-state of $\displaystyle m(t)$, that I guess, for the moment could be treated as a constant.

I know that the solution to the ODE is:

$\displaystyle m(t) = m(\infty) -\big(m(\infty)-m(0)\big ) \exp({-t/\tau} )$

being $\displaystyle m(0)$ the initial condition.

I've tried to solve the equation with a piece of paper, but get stuck in the process. Im principle, this equation should be easily solved by separation of variables...

What I tried so far:
First, separation of variables

$\displaystyle \frac{dm(t)}{m(\infty) - m(t)} = \frac{dt}{\tau}$

applying the integration limits:

$\displaystyle \int\limits_{0}^{t} \frac{dm(t)}{m(\infty) - m(t)} =\int\limits_{0}^{t} \frac{dt}{\tau}$

solving the integral

$\displaystyle \log\frac{m(\infty) - m(t)}{m(\infty) - m(0)} =\frac{t}{\tau}$
This integration is wrong. Letting "$\displaystyle u= m(\infty)- m$" gives $\displaystyle du= -dm$. You have the sign wrong in the integral so the fraction in the logarith m is inverted. You should have:
$\displaystyle \log\left(\dfrac{m(\infty)- m(0)}{m(\infty)- m(t)}\right)= \dfrac{t}{\tau}$

and applying exponential to both sides of the equation leads to:

$\displaystyle \frac{m(\infty) - m(t)}{m(\infty) - m(0)} =\exp(t/\tau)$

organizing...
$\displaystyle m(\infty) - m(t) = \big(m(\infty) - m(0)\big) \exp(t/\tau)$

finally, solving for $\displaystyle m(t)$

$\displaystyle m(t) = m(\infty) - \big(m(\infty) - m(0)\big) \exp(t/\tau)$

which is almost the solution, but the exponential is raised to $\displaystyle t/\tau$ and not to $\displaystyle -t/\tau$ as it should be.

I would appreciate if somebody would point out where is the mistake in the calculation

Thanks

Jose

3. Re: ODE solution of exponential form

Fantastic! Thank you very much for your help!