Introduction to Differential Equations

Equations Involving Derivatives

A differential equation relates a function to its derivatives — for example $\frac{dy}{dx} = 2y$, which says "the rate of change of $y$ equals twice $y$." Differential equations are the language of change: they describe how quantities evolve, and solving one means finding the function that satisfies the relationship. They are arguably the most important application of calculus in science.

A solution is a function that satisfies the equation. Differential equations typically have a family of solutions (with a constant), reflecting the constant of integration; a specific initial condition (like $y(0) = 3$) pins down one particular solution. The order of a differential equation is that of the highest derivative appearing.

The simplest solvable type is a first-order separable equation, where the variables can be separated onto opposite sides: $$\frac{dy}{dx} = g(x)h(y) \;\Rightarrow\; \frac{dy}{h(y)} = g(x)\, dx.$$ Then integrate both sides to find the solution. For the growth equation $\frac{dy}{dx} = ky$: separate to $\frac{dy}{y} = k\, dx$, integrate to $\ln|y| = kx + C$, and exponentiate to $y = y_0 e^{kx}$ — the exponential growth/decay law, derived directly.

This connects to earlier units: exponential growth ($\frac{dN}{dt} = kN$), radioactive decay, Newton's law of cooling ($\frac{dT}{dt} = -k(T - T_{env})$), and simple population models are all separable first-order equations solved by this integrate-both-sides method. Differential equations are where calculus becomes the engine of modeling the physical world.

Common pitfall: forgetting that a differential equation has a family of solutions and that an initial condition is needed to select one — and dropping the constant of integration when solving. Integrating both sides of a separable equation introduces a $+C$ (the general solution is a family); only an initial condition fixes $C$ to give the one particular solution. Omitting $C$, or treating the general solution as a single function, loses the whole family.

A slope field of little accent tangent segments with one solution curve threading through them from an initial point, following the prescribed dy/dx at each location.

$$\frac{dy}{dx} = ky \;\Rightarrow\; y = y_0 e^{kx}$$