Differential Equations Summary

Calculus 2 · Axiom Academy

SUMMARY Differential Equations Summary How differential equations turn a rule about rates of change into a function that predicts the future of a system. A differential equation relates a quantity to its own rate of change. Its solution is a function y(x) , not a single number. One initial condition pins down one curve. The general solution is a whole family; an initial value y(x_0)=y_0 selects the unique particular solution. Two go-to tools. When variables separate, integrate both sides; when they don't, step forward numerically with Euler's method. The exponential law y=y_0e^ kt is everywhere. It is the solution of , governing growth ( ) and decay ( ). Three canonical families recur. Exponential, logistic (bounded growth), and second-order linear (oscillation) cover most first-pass models. Core Concept Equations, Order & Initial Values A differential equation involves derivatives and describes how quantities change together. Its order is the highest derivative present, and its solution is a function — solving it means recovering y(x) from a statement about its slope. When to use: any time a problem gives you a rule for a rate rather than a value directly. Watch out for: the general solution has arbitrary constants; an initial condition y(x_0)=y_0 is needed to get the one particular solution.

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