Getting the general solution by using partial fractions [Differential Equation]

Hi all,

I'm not really confident with partial fractions, so this question really has me stumped at the moment.

For (factorized form of Constant Harvest/Constant Effort models):

How would I use variable separation and partial fractions to find the general solution?

So far I've made it to:

Am I on the right track?

P does not have to be the subject.

Thank you!

Re: Getting the general solution by using partial fractions [Differential Equation]

Quote:

Originally Posted by

**DVS** Hi all,

I'm not really confident with partial fractions, so this question really has me stumped at the moment.

For (factorized form of Constant Harvest/Constant Effort models):

How would I use variable separation and partial fractions to find the general solution?

So far I've made it to:

Am I on the right track?

P does not have to be the subject.

Thank you!

Are you using to represent or ? Same for Ps?

Re: Getting the general solution by using partial fractions [Differential Equation]

Hey DVS.

The first thing I recommend is to take P outside of both terms. Doing this will give you:

[1/P^2]*[1-u)^(-1)*[s-1]^(-1) which means you don't have to do a thing since s and u are independent variables of the one that is being integrating (i.e. you treat this as [1/P^2]*d where d is a constant.

So the integral of d*[1/P^2] is just -d/P + C for an indefinite integral but you can carry this to the RHS and put the C there.

Re: Getting the general solution by using partial fractions [Differential Equation]

Quote:

Originally Posted by

**Prove It** Are you using

to represent

or

? Same for Ps?

Apologies for not mentioning that.

is the unstable equilibrium

is the stable equilibrium

So it would be , not

Re: Getting the general solution by using partial fractions [Differential Equation]

You just have to rewrite 1/((P-Pu)(Ps-P)) on the form = (A/(P-Pu))+(B/(P-Ps)) with A, B to be expressed as functions on Pu and Ps.

Re: Getting the general solution by using partial fractions [Differential Equation]

Had another look at it earlier,

I now have:

= +

Then as I believe, I am supposed to multiply both sides by the denominator of the left side, which gives me:

But I don't see what that gets me or where I go to find the general solution from here. (Worried)

Re: Getting the general solution by using partial fractions [Differential Equation]

First, simplify. Then expand.

Re: Getting the general solution by using partial fractions [Differential Equation]

Now:

But I still don't understand - how does expanding that help? :confused:

Re: Getting the general solution by using partial fractions [Differential Equation]

1 = B*P - A*P +A*Ps -B*Pu

What is the relationship between B and A so that the expression be constant for any value of P ?

Re: Getting the general solution by using partial fractions [Differential Equation]