Introduction Summary

Differential Equations · Axiom Academy

SUMMARY Introduction to Differential Equations A differential equation is an equation about rates of change. This unit built the vocabulary to classify them, the skill to verify and visualize their solutions, and the theory that tells us when a solution is the only one. A differential equation relates a function to its own derivatives — we solve for an entire function, not a single number. Order and linearity classify an equation and determine which solution techniques apply. Direction fields visualize every solution at once — a small line segment of slope f(x,y) at each point, which every solution curve must stay tangent to. An initial condition picks one curve out of an infinite family , turning a general solution into a particular one. Existence and uniqueness is a genuine theorem with real conditions — it can fail, and when it does, an IVP can have more than one solution. Core Concept What Is a Differential Equation? An equation involving an unknown function and one or more of its derivatives. Instead of solving for a number, we solve for a function that satisfies a relationship between itself and its own rate of change — statements like "temperature decreases proportionally to how far it is from room temperature" translate directly into one. When to use: any process described by a rate of change — growth, decay, cooling, motion, circuits. Watch out for: the solution is a function, y(x) , not a single value. Core Concept Classifying by Order & Linearity

This is the written version of the interactive lesson above. See the full Differential Equations course.