Solving a differential equation
By Anonymous on Thursday, November 9, 2000 - 09:15 pm :

Hello,

I have to solve the following differential equation:

dv/dt + 0.000856v2 = -F/120000 where F is a positive constant.

I'm not sure how to do it.
What is the auxiliary equation for this system?
What is my trial function in order to find the particular integral?

Thank you very much,

By William Astle (Wja24) on Thursday, November 9, 2000 - 09:40 pm :
Here is one method.

You need two techniques:

  1. Integration by separation of variables, that is if dy/dx=f(x) then òdy=òf(x) dx
  2. The ability to solve partial fractions.
I will put C2 = 0.000856 and K2 = -F/120000 to make things simpler to read. You want to solve

dv/dt + C2 v2 = K2.

First we rearrange the equation.

dv/dt = K2 - C2 v2

so:

dt/dv = 1/(K2 - C2 v2 )

Separate the variables:

òdt = ò1/(K2 - C2 v2 )dv

Now we express

1/(K2 -C2 v2 ) as the sum of two fractions thus:

A/(K-Cv) + B/(K+CV)

If you multiply these fractions up and equate the result we see that A = B and A = (1/(2K))

So

t = ò(1/[(2K)(K-Cv)] + 1/[(2K)(K+CV)]) dv

You can do this integrations and get a couple of log terms, then its just a matter of using the properties of logs and rearranging to get a result for v in terms of t.