# How to solve this differential equation?

• Feb 6th 2013, 09:00 PM
joeschmo
How to solve this differential equation?
Hi all,

I'm trying to (re-)teach myself differential equations, and I've run into a snag. Say I have a differential equation which includes the solution on the RHS. How can I solve this, I think I can put it into standard form, but then I get stuck. Say I have

$\displaystyle $$\frac{dP}{dt}=.3P-20$$$

and where

$\displaystyle $$P(0)=35$$$

where P is the function I'm trying to obtain. I would think I could put this in standard form like so

$\displaystyle $$\frac{dP}{dt}-.3P=-20$$$

with an integrating factor of

$\displaystyle $$v(t)=\exp{(-\int.3dt)}=\exp(-.3t)$$$

So I'd get

$\displaystyle $$\exp{(-.3t)}(\frac{dP}{dt}-.3P)=-20\exp{(-.3t)}$$$

I know that the solution is the following, but I'm not sure of how to get there.

$\displaystyle $$P(t)=(20-20\exp{(.3t)}+.3(35)\exp{(.3t)}/.3)$$$

Any help would be appreciated.
• Feb 6th 2013, 09:08 PM
chiro
Re: How to solve this differential equation?
Hey joeschmo.

You should probably look at this:

Integrating factor - Wikipedia, the free encyclopedia

Note that you can also use the fact that dP/(0.3P - 20) = dt and integrate both sides to get the same answer as you would obtain from using
integrating factors.
• Feb 7th 2013, 04:46 AM
joeschmo
Re: How to solve this differential equation?
Ah, that does it. I guess I can solve this by separation of variables. Thanks for the hint. Still didn't get there with the integrating factor yet but I'll give it another crack later.