Differential Equations

The unknown is a function rather than a number, so an answer is a whole curve: order, linearity, the constants a general solution carries, and how a direction field draws the family before anything has been solved.

When the right side factors into a function of {t} times a function of {y}, both sides can be integrated: decay, cooling, the logistic curve, and the constant solutions that dividing quietly destroys.

Multiplying by {{e}}^{{\int }{p}{d}{t}} forces the left side to be an exact derivative, which solves every first order linear equation and splits its answer into a transient and a steady state.

Picard's theorem says when an initial value problem has exactly one solution, its two failures are blow-up and a bucket that cannot say when it started draining, and Euler and Runge-Kutta produce numbers where no formula exists.

When the right side does not depend on time, the sign of {f}{(}{y}{)} alone fixes the long-term behaviour: the phase line, the linearisation test {f}{\text{'}}{(}{{y}}^{{*}}{)}, terminal velocity, and a harvested fishery whose stable state collides with an unstable one and vanishes.

Newton's second law is second order, so the theory is rebuilt: the linear operator, superposition, the proof that the solution space has dimension exactly two, the Wronskian test for independence, and reduction of order.

Substituting {{e}}^{{r}{t}} turns the differential equation into a quadratic: two real roots, a repeated root recovered by reduction of order, and complex roots turned back into real decaying oscillations by Euler's formula.

One equation for a mass on a spring and an RLC circuit: the natural frequency, the damping ratio that classifies every case, why a car and a door closer are tuned to opposite sides of critical, and how to measure damping from two peaks.

Undetermined coefficients gives the steady state of a driven oscillator, and its amplitude and phase as functions of driving frequency show a peak that grows like {1}{/}{2}{\zeta }, unbounded growth like {t} without damping, and why the famous bridge failure was not resonance.

Variation of parameters produces a particular solution for any forcing at all, and reading the result as an integral shows that the impulse response, the step response and the frequency response are one object seen three ways.

Every equation of any order is a first order system: coupled tanks, the trial {{e}}^{{\lambda }{t}} that gives a characteristic equation in trace and determinant, and the sorting of every linear plane flow into node, saddle, spiral or centre.

Linearising at each equilibrium gives the local picture and conserved energy gives the global one: the pendulum's separatrix and its period growing with amplitude, predator-prey cycles that a small change destroys, limit cycles, and where prediction stops.