Unit 7: Differential Equations
Work through these 9 topics in order. Clear the gatekeeper and the rest of the course opens up.
7.1 Modeling situations with differential equations
A differential equation is a sentence about a RATE.
7.2 Verifying solutions for differential equations
A proposed solution is just a candidate function.
7.3 Sketching slope fields
A slope field is a picture of a differential equation: at a grid of points, draw a tiny segment whose slope is what \dfrac{dy}{dx} evaluates to there.
7.4 Reasoning using slope fields
Given a slope field, you can describe solutions without a formula: where they rise or fall, where equilibria sit, and roughly what a particular solution through a given point looks like.
7.5 Approximating solutions using Euler's method (BC)
When you cannot solve a DE exactly, march along it in small steps.
7.6 Finding general solutions using separation of variables
If a DE can be rearranged so all the y’s (with dy) sit on one side and all the x’s (with dx) on the other, you can integrate each side separately to recover y.
7.7 Finding particular solutions using initial conditions
The general solution has a free constant — a whole family of curves.
7.8 Exponential models with differential equations
Whenever a quantity changes at a rate proportional to itself, it grows or decays exponentially.
7.9 Logistic models with differential equations (BC)
Real populations cannot grow forever.