# Help with a simple first order differential equation that seems to contradict itself.

• Feb 23rd 2011, 12:48 PM
Yehia
Help with a simple first order differential equation that seems to contradict itself.
So given a differential equation that models the mass of a substance (m) produced during a chemical reaction after time (t).

equation: dm/dt = (50-m)²/500 (given m=0 when t=0)

after i solved it, i get: m = (50) + (500/(t-10))

but this looks problematic.
firstly it implies that the substance keeps getting smaller mass, shouldn't m be getting bigger as t does?!?!?!?

secondly, during the first 10 seconds the mass is negative!

and also t can never equal 10 then!!! (denominator will be 0).

and lastly, i'm asked to show that the mass will never exceed 50...but wait isn't it ALWAYS going to be bigger than 50?!?! it never gets below 50!.

• Feb 23rd 2011, 01:00 PM
e^(i*pi)
You might have got a couple of sign errors in there because I get an answer different in sign. I have put my working in spoiler

Spoiler:
$\displaystyle \dfrac{dm}{(50-m)^2} = \dfrac{dt}{500}$

$\displaystyle u = 50-m \rightarrow du = -dm \implies dm = -du$

Giving $\displaystyle -\dfrac{du}{u^2} = \dfrac{dt}{500}$

$\displaystyle \dfrac{1}{u} = \dfrac{1}{50-m} = \dfrac{t}{500} + C$

Using your initial condition $\displaystyle C = \dfrac{10}{500}$

$\displaystyle 1 = (50-m) \left(\dfrac{t+10}{500}\right)$

$\displaystyle 50-m = \dfrac{500}{t+10}$

$\displaystyle m = 50 - \dfrac{500}{t+10}$

What do you know about $\displaystyle t+10$ and it's relationship to 0?
• Feb 23rd 2011, 01:10 PM
Yehia
Quote:

Originally Posted by e^(i*pi)
You might have got a couple of sign errors in there because I get an answer different in sign. I have put my working in spoiler

Spoiler:
$\displaystyle \dfrac{dm}{(50-m)^2} = \dfrac{dt}{500}$

$\displaystyle u = 50-m \rightarrow du = -dm \implies dm = -du$

Giving $\displaystyle -\dfrac{du}{u^2} = \dfrac{dt}{500}$

$\displaystyle \dfrac{1}{u} = \dfrac{1}{50-m} = \dfrac{t}{500} + C$

Using your initial condition $\displaystyle C = \dfrac{10}{500}$

$\displaystyle 1 = (50-m) \left(\dfrac{t+10}{500}\right)$

$\displaystyle 50-m = \dfrac{500}{t+10}$

$\displaystyle m = 50 - \dfrac{500}{t+10}$

What do you know about $\displaystyle t+10$ and it's relationship to 0?

*sigh*

Thank you so much, i mistakenly forgot to multiply the (-1/50-m) by -1, so as you said, i got my signs mixed up.
that clarifies everything and clearly t can never be -10 seconds, so the denominator will never be 0.
and clearly as t tends towards infinite, m tends towards 50 (but never reaches it) which explains my quiery.

i really want to kick myself for spending ages on something that was so obvious!