Solve ODE: Integrating Factor Solution w/ Mike's Query

[bsodmike]

Homework Statement

This is a solution detailed in a past paper,

http://stuff.bsodmike.com/sensors_pastpaper.png

Homework Equations

Utilise an integrating factor to solve as detailed by myself https://www.physicsforums.com/showthread.php?t=283610".

The Attempt at a Solution

It can be said that,

\dfrac{d\theta}{dt}+\dfrac{\theta}{\beta}=\dfrac{\theta_m}{\beta}

Hence, the solution by employing an integrating factor would yield,

e^{t/\beta}\theta=\dfrac{\theta_m}{\beta}\int{e^{t/\beta}\,dt}

\theta=\dfrac{\theta_m}{\beta}+Ce^{-t/\beta}

The problem is that I'm having the extra 'beta' term as shown above. Any ideas?

Thanks!
Mike

 

[bsodmike]

A more direct approach, solving the homogenous part:

\int{\dfrac{1}{\theta}\,d\theta}=\int{-\dfrac{1}{\beta}\,dt}

ln(\theta)=-\dfrac{t}{\beta}+C

\theta=e^{-t/\beta+C}

Now, my memory here is fuzzy but e^{-t/\beta+C} = e^{-t/\beta}+e^C ??

 

[bsodmike]

Ah, another small blunder, of course!

Now, \int{e^{t/\beta}\,dt}=\beta e^{t/\beta}+C. thus, the 'beta's cancel,

e^{t/\beta}\theta=\theta_m e^{t/\beta}+C

Dividing by e^{t/\beta}, yields,

\theta=Ce^{-t/\beta}+\theta_m