First Order ODE

• Sep 29th 2011, 05:49 PM
joatmon
First Order ODE
$\displaystyle \frac{dz}{dt} + 3e^{t+z} = 0$

Here's what I have done:

$\displaystyle \frac{dz}{dt} = -3e^{t+z}$

$\displaystyle \ln(dz) - \ln(dt) = \ln(-3) + ln(e^{t+z})$

Am I approaching this correctly? Something is obviously wrong since the ln(-3) is undefined, which makes me think that I am doing this wrong.

Thanks!
• Sep 29th 2011, 06:00 PM
pickslides
re: First Order ODE
I see it like this

$\displaystyle \frac{dz}{dt} = -3e^{t+z}$

$\displaystyle \frac{dz}{dt} = -3e^{t}e^{z}$

$\displaystyle e^{-z}~dz= -3e^{t}~dt$

$\displaystyle \int e^{-z}~dz= -3 \int e^{t}~dt$

etc..
• Sep 29th 2011, 06:13 PM
joatmon
re: First Order ODE
Thanks for that. This is my very first differential equation (my deflowering, so to speak), so this is all new to me.
• Sep 29th 2011, 06:19 PM
joatmon
re: First Order ODE
But wait...

Carrying on from where you left off:

$\displaystyle e^{-z} = -3e^t + C$

$\displaystyle -z = \ln(-3e^t + C)$

I'm back to my same dilemma. I have the logarithm of a negative number. How do I handle that? Thanks again.
• Sep 29th 2011, 06:44 PM
pickslides
re: First Order ODE
Quote:

Originally Posted by joatmon

$\displaystyle -z = \ln(-3e^t + C)$

I'm back to my same dilemma. I have the logarithm of a negative number. How do I handle that?

But do you? consider the case where t=0 and C=4. Is the logarithm negative?
• Sep 29th 2011, 06:50 PM
joatmon
re: First Order ODE
I thought of that, too, that the constant can always push the equation positive. Where I am confused is that when I run this problem through Wolfram (or my calculator) to check my work, it shows the answer as $\displaystyle \ln(3e^t + C)$. I'm just trying to understand whether I am getting the right answer or not, and I have two sources (that I trust more than my own work) that have a slightly different answer.

Any thoughts? Thanks again.
• Sep 30th 2011, 06:45 AM
Ackbeet
Re: First Order ODE
Quote:

Originally Posted by joatmon
But wait...

Carrying on from where you left off:

$\displaystyle e^{-z} = -3e^t + C$

$\displaystyle -z = \ln(-3e^t + C)$

I'm back to my same dilemma. I have the logarithm of a negative number. How do I handle that? Thanks again.

I don't quite get the same result. I have

$\displaystyle -e^{-z}=-3e^{t}+C,$ and so

$\displaystyle e^{-z}=3e^{t}+C,$ where I've redefined my constant. Then

$\displaystyle -z=\ln(3e^{t}+C),$ and hence

$\displaystyle z=-\ln(3e^{t}+C).$