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.
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.
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.