Differential equations
Many laws say only how fast something changes. How do you turn such a law into an equation, and what can you learn about the function that obeys it?
Where you are 10 inside
What this area is about
A differential equation relates an unknown function to its own derivatives, as \displaystyle \frac{dy}{dt} = ky does. Its solutions are functions, not numbers. Some can be found by integrating; for the rest, pictures and step-by-step estimates still show how the solutions behave.
How these fit together
Setting up. Writing a differential equation turns a sentence about a rate into an equation, and differential equations and their solutions tests a proposed solution by substituting it.
Seeing and estimating. Slope fields draw the slope the equation gives at each point. Euler’s method follows those slopes in small steps to estimate a solution.
Solving. Separable equations are undone by integrating both sides, then fixing the constant from a starting value.
Models. Exponential growth and decay solves y' = ky and Newton’s law of cooling, T' = k(T - A), and logistic growth adds a ceiling the population approaches.
Beyond a first course. Three college areas, not yet written: first-order equations (slope fields, separable and linear equations, and when solutions exist), second-order linear equations (constant-coefficient equations, forcing, and oscillation) and systems and transforms (systems, phase portraits, and the Laplace and Fourier methods).
A path through it
Writing an equation, then checking a solution, then slope fields, which Euler’s method and separable equations both need. Exponential models follow separable equations, and logistic growth follows exponential models, also using partial fractions and concavity. Euler’s method can come last, since its examples reuse those models.
The college areas build on slope fields and separable equations, in the order first-order, second-order, systems.