Differential equations
Equations where the unknown is a function — and a derivative is in there too.
A differential equation relates a function to its derivatives. Solutions are *functions*, not numbers. They model populations, cooling, motion, electric circuits — anything that changes.
Solutions are functions, and there are many
Algebra equations have number answers; differential equations have *function* answers — and usually a whole family of them, one per value of the constant C. Slide C above: every curve solves dy/dx = 0.3y. Pin down C with an initial condition like y(0) = 2, and you get the one solution that fits your situation.
Vocabulary that gets thrown around
- Order — the highest derivative present (dy/dx is first order, d²y/dx² is second).
- Linear — y and its derivatives appear only to the first power, not multiplied together.
- Ordinary (ODE) vs partial (PDE) — one independent variable vs several.
- General solution (with the C's) vs particular solution (constants fixed by initial/boundary conditions).
Newton's second law F = m·d²x/dt², radioactive decay, population growth, Newton's law of cooling, RC circuits, the heat equation — physical laws are very often differential equations.
Verify that y = Ce^(0.3x) solves dy/dx = 0.3y.
dy/dx = y
Functions whose derivative is themselves. Answer: y = Ce^x. Try it: d/dx(Ce^x) = Ce^x. ✓
Why they matter
Newton's second law (F = ma = m · d²x/dt²) is a differential equation. Almost every physical law is one.