I have a vague memory of going over these types of integrals, but have yet to fully visualize and comprehend the patterns that they imply. I'm now stuck on a problem because the answer is different from mine and the solution guide skips it. Any help you can give to help me with the concept and/or the problem would be appreciated.
What I think I understand about differential equations:
A differential equation is an equation that relates a formula and it's derivative in the same equation. So that often means that a differential equation is in the for of dy/dx=f(y) where y=f(x), right? I've understood direction fields of differential equations and can piece together their pattern of divergence versus convergence and to what value they are diverging or converging to.
My Conceptual Questions:
1. Does the relationship between a differential equation and a direction field mean that a differential equation dy/dx=f(y) is always a pattern that tends to generalize the behavior of many different functions y=f(x)?
2. Does every y=f(x) have at least one dy/dx=f(y) that could describe it? If so, are there ways to determine the later from the former? If not, what sorts of situations tend to call for the later?
3. On a direction field, when a curve made from the field reaches a value y, is it's curve from that point onward identical (super imposable?) with any other curve drawn from the direction field from that same value of y onwards?
The Problem, My answer, and the answer at the end of the book:
Consider the field mice population equation in Ex1 [An equation for the population of field mice trends with a given owl population preying on them] dp/dt=0.5p-450. Find the time at which the population becomes extinct if p(0)=850. [t is in months]
divide both sides by 1/2
integrate both sides
t=2ln18 approx=5.78 months
Any and all help is appreciated