Differential Equations
A differential equation relates a function to its own rates of change. You rarely get the function handed to you; instead you get a law it must obey at every instant, and the work is recovering the function from that law.
The first thing to read off an equation is its geometry. A first-order law assigns a slope to every point of the plane, and its solutions trace the curves that stay tangent to that field everywhere, so you can see their shape long before you can write a formula.
For linear systems the geometry becomes a phase portrait. The eigenvalues of the coefficient matrix decide everything — real and negative gives decay, imaginary gives rotation, positive real part sends trajectories away — and the whole flow sorts into nodes, saddles, and spirals.
Exact methods come next: separation, integrating factors, and characteristic equations solve the first- and second-order equations outright, while series and the Laplace transform reach the ones that resist elementary functions.
The second-order linear equation is the workhorse of physics: a mass on a spring, a circuit, a swinging pendulum. With damping its solution is an oscillation trapped inside a shrinking exponential envelope — the balance of restoring force and dissipation made visible.
Underneath all of it sits the simplest equation, y' = k y, whose solution is a bare exponential. Growth, decay, cooling, and half-lives are all this one law with a sign chosen; every richer model is measured against it.
Most equations that matter are nonlinear, and there the goal shifts from formulas to behaviour — equilibria, stability, limit cycles, and the onset of chaos, questions you answer qualitatively even when no closed form exists.
When even that fails you integrate numerically, stepping the solution forward in small increments; and Fourier methods close the loop, turning the classical partial differential equations of heat, waves, and potential back into ordinary ones.