First-Order Ordinary Differential Equations

It means finding a function $y(x)$ such that plugging $y$ and $y'$ into the equation makes it an identity for all $x$ in some interval. The equation tells you the slope at every point; the solution is a curve whose tangent at each point matches that prescribed slope. There are typically infinitely many solutions — a one-parameter family — and an initial condition $y(x_0)=y_0$ picks out one specific solution. Geometrically, the slope field shows the prescribed slopes; integral curves "thread through" the field tangentially.

The slope field draws short tangent segments at a regular grid of points. At each $(x,y)$, the segment's slope is exactly $f(x,y)$ — what $y'$ must be there if a solution passes through. The integral curve is computed by RK4 starting at your chosen $(x_0, y_0)$: at each step it samples the slope at four nearby points and takes a weighted average to produce the next point. The result is a curve that's tangent to the slope field at every step — i.e., a solution of the ODE — accurate to $O(h^4)$ per step.

Whenever a quantity changes at a rate that depends on its current value or on time. Population growth: $dP/dt = rP$ → exponential, then logistic when resources cap. Newton's cooling: a hot object cools at rate proportional to temperature gap with surroundings — first-order linear. Radioactive decay: $dN/dt = -\lambda N$. Drug pharmacokinetics: drug concentration in bloodstream obeys first-order elimination + dosing input. RC circuits: voltage across capacitor satisfies $RC\,V' + V = V_s(t)$ — exactly the linear ODE we solved. Compound interest with continuous deposits: linear ODE for account balance.

If two solutions $y_1$ and $y_2$ crossed at a point $(x^*, y^*)$, both would satisfy $y(x^*) = y^*$ and $y'(x^*) = f(x^*, y^*)$ — same initial value, same slope. By Picard's uniqueness theorem (assuming Lipschitz $f$), there's only one solution through any point — so $y_1 = y_2$ in a neighbourhood, contradicting the assumption they were different curves. The slope field is a "non-crossing flow". When uniqueness fails (e.g., $y' = 3y^{2/3}$ at $y=0$ — not Lipschitz there), curves can branch: $y \equiv 0$ and $y = (x-c)^3$ both satisfy $y(c)=0$.

RK4 takes the step $y_{n+1} = y_n + \tfrac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)$ where $k_1 = f(x_n, y_n)$, $k_2 = f(x_n + h/2,\ y_n + h k_1/2)$, $k_3 = f(x_n + h/2,\ y_n + h k_2/2)$, $k_4 = f(x_n + h,\ y_n + h k_3)$. The four-point Simpson-like weighted average matches the Taylor series of the true solution up to and including $h^4$ terms — the leading error is $O(h^5)$ per step, accumulating to $O(h^4)$ globally over a fixed interval. Cost per step: 4 function evaluations vs 1 for Euler. To gain a factor of 10 in accuracy: Euler needs $10\times$ more steps, RK4 needs only $10^{1/4}\approx 1.78\times$ more — RK4 wins enormously for tight tolerances.

An autonomous ODE $y' = f(y)$ (no $x$ dependence) defines a vector field on the line/phase space — at each $y$, a velocity vector $f(y)$. Its solution is the flow generated by integrating that field. Equilibria $f(y^*)=0$ are fixed points; their stability is determined by $f'(y^*)$: stable if $f'(y^*) < 0$, unstable if $> 0$. Generalize to $\R^n$: $\mathbf{y}' = \mathbf{F}(\mathbf{y})$ — autonomous system. Equilibria of nonlinear systems are classified by eigenvalues of the Jacobian (linearization theorem of Hartman-Grobman). This is the start of dynamical systems theory: bifurcations, attractors, chaos, structural stability — all built on the geometric view of ODEs as flows of vector fields.

They're three orthogonal "structures" that allow closed-form integration. Separable: $y'$ factors as $g(x)h(y)$ — variables disentangle, two independent integrals do the job. Linear: $y$ and $y'$ appear only to the first power, no products of $y$ with itself — integrating factor turns the LHS into a perfect derivative. Exact: the ODE rewritten as $M\,dx + N\,dy = 0$ is the differential of some $F$ — the gradient/conservation viewpoint. Most random ODEs aren't any of these (e.g.\ $y' = \cos(xy)$). Recognizing the form is the first triage step before reaching for numerics. There are also tricks (substitutions: homogeneous, Bernoulli, Riccati) that convert a non-classical equation to one of these three.