Solving Logistic Differential Equation, Cover up for partial fractions (why and how it works): • the cover-up method & ... For more calculus 2 tutorials: / justcalculus
1:53 I should point out, that I use your trick here to avoid partial fractions: factor out another P in the denominator, so you have M/(P^2 * (MP^(-1) - 1)) and then bring the P^2 to the numerator as P^(-2), then let u= the denominator.
man, isn't this just beautiful? and such a clean procedure. although I kind of wanted to see that substitution of c (in terms of the initial population) made
hello! can you please do also the Logistic Differential Equation With Allee Effect? We badly need it for our special problem set in differential equation but we are having a hard time to derive the equation. Thank you in advance!
You didn't need that absolute value in P/(M-P) because P is +ve and M is the carrying capacity, so P is always less than M. Therefore the whole term becomes +ve. Anyway, nice video as always.
The starting population can be more than the carrying capacity, and the population will decrease to meet the carrying capacity. You can see it if you make a slope field.
that was indeed legitimate, because we're talking about a completely arbitrary constant, eg. literally anything that is not dependent on x. You can just think of C2 as absolute value of C3 and then it makes complete sense. We can do many weird things with constants in diff equations, because in the end you can propagate all the steps backwards and you will see it's all legitimate, because every step in between was also legitimate. (eg. if one of the steps involved a logarithm, then the constant will cause the expression it equals to still be positive, even if you plugged a negative number at the start)
He talked through it, and they were simple enough that both constants ended up becoming one. He uses Heaviside cover-up whenever possible, and on this one, it was possible for both.
@@victoria673 P=0 is one of the stationary points of the logistic equation, in addition to P=M. Any solution should work and make mathematical sense at both the points. If it is logic that you root for, then the term ln(|p|) also has no meaning different from ln(p) as p cannot be negative. My point was the value of the constant 'c' in the final solution. c = (M-p0)/p0. What if p0=0? Is it a positive or a negative quantity??
@@elliottmanley5182 This considers the pretense that the species already exists, and that the parameters that govern its growth are constant, such as its fertility rate, death rate, and the capacity of its environment. This is only an approximation to reality, since it doesn't consider interaction with other species, changes in its environment, or evolution. At one point, obviously the species had to originate, when the population technically was zero, but that's a topic for another subject.
@@tcwan0501 What did you say to him? I could translate his response and your second comment, but I can't find anything on what that character means in your original post.
