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!
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.
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.
You want the two polynomials and to be equal. That means that all the coefficients of have to be equal. To figure that out, you need to expand :
Which has to equal . That means that and . This is two linear equations in the two unknowns and , so you can solve for them. The answer ends up being pretty intuitive - not always the case with partial fractions. I usually like to double-check my final partial fraction result before moving on.
There's another way to figure out what and are, which is based on the principle that if two polynomials in P are equal, then they have to be equal for all values of P.
- Hollywood