Differential Equations

Formulas

Differential equations are the language in which nature describes change. When a physicist writes Newton’s second law F = ma, they are essentially writing a differential equation: acceleration a = x'' is the second derivative of position x with respect to time. The heat conduction equation, Maxwe...

The history of differential equations is inseparable from the history of physics. Newton created mathematical analysis precisely to solve mechanics problems—and the first differential equations in the history of science were equations describing the motion of planets. Since then, the field has sp...

ODE (ordinary differential equation) links a function of one variable y(x) to its derivatives: F(x, y, y', y'', ..., y^(n)) = 0. The word "ordinary" distinguishes them from partial differential equations, where the function depends on several variables.

The order of an ODE is the highest order of derivative that appears in the equation. The equation y' = ky is first order; y'' + ω²y = 0 (harmonic oscillator) is second order.

Formulas

If the function $F(x, y)$ is differentiable, then its total differential is $dF = (\partial F/\partial x)\,dx + (\partial F/\partial y)\,dy$. The equation $dF = 0$ means that $F$ is constant along the integral curves: $F(x, y) = C$. This is the implicit solution—the family of level lines of the f...

But what if we are given the equation $P(x,y)\,dx + Q(x,y)\,dy = 0$, and do not know if it is the differential of some function? The criterion for exactness and the method for finding $F$—this is the theory of exact equations.

The equation $P(x,y)\,dx + Q(x,y)\,dy = 0$ is called exact if there exists a function $F(x,y)$ such that $\partial F/\partial x = P$ and $\partial F/\partial y = Q$. Then the equation takes the form $dF = 0$, and its general solution is $F(x, y) = C$.

Criterion of exactness: If $P$ and $Q$ have continuous partial derivatives in a simply connected domain, then the equation is exact if and only if $\partial P/\partial y = \partial Q/\partial x$.

Formulas

When solving an ODE $y' = f(x, y)$, we usually find the general solution—a family of curves depending on a constant $C$. It seems that by selecting the appropriate $C$, any solution can be obtained. But this is not always the case.

A singular solution is a solution that cannot be obtained from the general solution for any specific value of the constant $C$. It cannot be "fit" into the family, though it satisfies the equation. Such solutions have fundamental physical and geometric significance.

Geometrically, a singular solution is the envelope of a family of integral curves: a curve that, at each of its points, is tangent to at least one integral curve from the general family. At the points of tangency, a branch "splits off" from the general solution—the singular solution.

$p$-discriminant Method: Suppose the general solution is written as $F(x, y, C) = 0$. We seek singular solutions by eliminating $C$ from the system:

Formulas

Before solving a differential equation, one must ensure that a solution actually exists. And if it exists—is it unique? These questions are not merely academic: the answer determines whether it is even possible to predict the behavior of a system.

Let us consider a physical analogy. You describe the trajectory of a particle with the equation $x'' = F(x)/m$. If the velocity $x'$ at a given point in space is specified—is the future trajectory unique? If yes, then "Laplace determinism" holds: knowing the state of the system at time $t_0$, one...

Ordinary continuity of $f(x, y)$ turns out to be insufficient for uniqueness. A stronger condition is needed.

A function $f(x, y)$ satisfies the Lipschitz condition in $y$ in a domain $D$ with constant $L$ if for all $(x, y_1)$, $(x, y_2) \in D$:

Definitions

Lorenz system (1963)

— the simplest example of deterministic chaos:

In real-world problems, initial conditions are never known exactly. The position of a satellite is measured with an accuracy of one meter. The initial concentration of a substance in a reaction is determined with an accuracy of one percent. The air temperature at the launch of a meteorological mo...

The answer depends on the equation. Sometimes, the errors are small and manageable. Sometimes, they grow exponentially—and long-term forecasting becomes impossible.

If $f$ satisfies the Lipschitz condition with constant $L$, then solutions with close initial conditions remain close on a compact interval:

Estimate: $|y(x; y_0) - y(x; \tilde{y}_0)| \leq |y_0 - \tilde{y}_0| \cdot e^{L|x - x_0|}$.

Formulas

Picard's theorem requires the Lipschitz condition. What if this requirement is violated? Peano's theorem answers the question of existence in a more general case.

Peano's Theorem: If $f(x, y)$ is continuous in the rectangle $R = \{|x − x_0| \leq a, |y − y_0| \leq b\}$, then the Cauchy problem $y' = f(x, y),\ y(x_0) = y_0$ has at least one solution on $|x − x_0| \leq h = \min(a, b/M)$.

The key difference from Picard's theorem: only existence is asserted, but not uniqueness. Without the Lipschitz condition, several solutions may pass through a single point.

Proof uses the Arzelà–Ascoli theorem: from a sequence of approximations, an equicontinuous subsequence is selected, which converges (compactness in the space of continuous functions).

Formulas

Newton's law for a damped spring pendulum: $m\ddot{x} + c\dot{x} + kx = F(t)$. This is a second-order linear ODE with constant coefficients. The parameters $m$ (mass), $c$ (damping coefficient), and $k$ (spring stiffness) fully determine the system's behavior. Three qualitatively different regime...

A linear ODE of order $n$: $ L[y] = y^{(n)} + p_{n-1}(x) y^{(n-1)} + ... + p_{1}(x) y' + p_{0}(x) y = f(x). $

Key theorem on the general solution: The general solution of the nonhomogeneous equation takes the form $y = y_p + y_g$, where $y_p$ is any particular solution of the nonhomogeneous equation, and $y_g$ is the general solution of the corresponding homogeneous equation $L[y] = 0$.

It follows that two problems must be solved: (1) find the solution space of the homogeneous equation; (2) find at least one particular solution of the nonhomogeneous equation.

We know how to find the general solution of the homogeneous equation. But real systems must respond to external influences—F(t) in the pendulum equation, EMF in the electric circuit equation, input signal in a control system. This makes the equation nonhomogeneous: L[y] = f(x). The method of vari...

The name "variation of constants" reflects the idea: in the general solution of the homogeneous equation c₁y₁ + c₂y₂ + ... + cₙyₙ, we replace the constants cᵢ by functions cᵢ(x)—"varying" them so as to satisfy the nonhomogeneous equation.

Let {y₁, ..., yₙ} be the fundamental system of solutions of the homogeneous equation. We seek a particular solution:

c₁' y₁ + ... + cₙ' yₙ = 0 c₁' y₁' + ... + cₙ' yₙ' = 0 ... c₁' y₁^{(n-2)} + ... + cₙ' yₙ^{(n-2)} = 0

Formulas

Euler equation: $x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \ldots + a_1 x y' + a_0 y = f(x)$.

Feature: the coefficients of $y^{(k)}$ are proportional to $x^k$. This is not accidental—such equations arise when solving problems with natural radial symmetry (polar, cylindrical, spherical coordinates), as well as in the search for power solutions of more general equations.

Substitution: $t = \ln x$ (for $x > 0$), that is, $x = e^t$. Let $D = \frac{d}{dt}$.

Key formulas: $x \frac{dy}{dx} = D y$, $x^2 \frac{d^2y}{dx^2} = D(D-1) y$, $x^3 \frac{d^3y}{dx^3} = D(D-1)(D-2) y$, and in general $x^k y^{(k)} = D(D-1)\cdots(D-k+1) y$.

Formulas

A single higher-order ODE $y^{(n)} = f(x, y, y', ..., y^{(n-1)})$ is equivalent to a first-order system. We introduce: $x_1 = y$, $x_2 = y'$, ..., $x_n = y^{(n-1)}$. Then $x_1' = x_2$, $x_2' = x_3$, ..., $x_n' = f(x, x_1, ..., x_n)$. This is the standard form of the system. Transitioning to matri...

In mechanics, systems of ODEs describe the motion of several interacting bodies. In electronics — current and voltage in complex circuits. In ecology — the joint dynamics of several populations (predator–prey model).

The system: $x_1' = a_{11}x_1 + ... + a_{1n}x_n + f_1$, ..., $x_n' = a_{n1}x_1 + ... + a_{nn}x_n + f_n$.

In vector form: x' = $A(t)$ x + f$(t)$, where x = $(x_1, ..., x_n)^\mathrm{T}$ is the column vector of state, $A$ is the $n\times n$ matrix of coefficients, f is the inhomogeneity vector.

Definitions

Pendulum (small oscillations): A = [[0, 1], [−ω₀², 0]]. tr A = 0; det A = ω₀² > 0. Center

— ideal undamped oscillations. (Friction converts the center into a stable focus.)

Formulas

When studying the system x' = Ax in ℝ², an explicit formula is not always necessary. Often, it is more important to understand the nature of the motion: does the system oscillate around equilibrium, tend toward it, or escape from it? For this purpose, a phase portrait is constructed—a family of t...

A phase portrait is a "map" of system behavior for all initial conditions at once. A single picture replaces an infinite number of separate graphs. It was precisely phase portraits that enabled Poincaré at the end of the 19th century to lay the foundations of the qualitative theory of differentia...

The behavior of trajectories near the equilibrium x* = 0 is completely determined by the eigenvalues λ₁, λ₂ of the matrix A.

Stable node: λ₁, λ₂ < 0 (real, negative). All trajectories tend to zero at exponential speed. The "fast" eigendirection (the more negative λ) dominates: trajectories are tangent to it as x → 0.

The world is predominantly nonlinear. A pendulum with large oscillations is described by the equation θ'' + (g/L) sin θ = 0—a nonlinear one. The predator–prey system follows the Lotka–Volterra equations—which are nonlinear. The Navier–Stokes equations of hydrodynamics are nonlinear. How do we ana...

The main instrument is linearization near an equilibrium point. The idea: in a small neighborhood of equilibrium, a nonlinear system “resembles” a linear one. We already know how to analyze the dynamics of linear systems via eigenvalues.

An equilibrium point (fixed point) of the system x' = f(x) is a point x*, where f(x*) = 0, that is, the derivative vanishes. At equilibrium, the system “stands” and does not change over time.

Example—the pendulum: θ'' = −(g/L) sin θ. In matrix form: x₁ = θ, x₂ = θ'. System: x₁' = x₂, x₂' = −(g/L) sin x₁. Equilibrium points: x₂ = 0 and sin x₁ = 0, that is, x₁ = nπ. Two types: θ = 0 (hanging pendulum) and θ = π (inverted pendulum).

Definitions

Formulation: Suppose V̇ ≤ 0 and the set M = {x

— V̇ = 0} contains no complete trajectories except x* = 0 itself. Then x* = 0 is asymptotically stable.

Formulas

In 1892, the young St. Petersburg mathematician Aleksandr Lyapunov defended his dissertation "The General Problem of the Stability of Motion", which revolutionized the theory of differential equations. Lyapunov proposed analyzing stability not by solving the equations explicitly, but by finding a...

An analogy from physics: a system is stable if its "energy" decreases along trajectories. Lyapunov generalized this observation to arbitrary dynamical systems by replacing actual energy with its mathematical analog—the Lyapunov function.

Consider the system x' = f(x, t), with f(0, t) = 0 (an equilibrium point at the origin).

Lyapunov Stability: The equilibrium x* = 0 is stable if for any ε > 0 there exists δ > 0 such that |x(t₀)| < δ implies |x(t)| < ε for all t ≥ t₀.

The eigenvalues of matrix A determine stability. But for a high-order matrix, calculating the roots of the characteristic polynomial is a labor-intensive task. In the 18th–19th centuries, mathematicians searched for algebraic criteria allowing one to check stability without explicitly finding the...

Historical problem: A steam engine regulator designer needs to know whether the engine will be stable for given parameters. He cannot solve a sixth-order polynomial equation. The Routh (1877) and Hurwitz (1895) criteria provide an answer in the form of several inequalities for the coefficients of...

For the polynomial $P(\lambda) = \lambda^n + a_{n-1} \lambda^{n-1} + \dots + a_1 \lambda + a_0$, we construct the Hurwitz matrix:

$ H_n = \begin{bmatrix} a_{n-1} & a_{n-3} & a_{n-5} & \dots \\ 1 & a_{n-2} & a_{n-4} & \dots \\ 0 & a_{n-1} & a_{n-3} & \dots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix} $

The word "bifurcation" (from Latin bifurcus — forked) in mathematics means a qualitative change in the behavior of a dynamical system with a small change in a parameter. Before the bifurcation — one picture of behavior, after — fundamentally different.

Bifurcations are a mathematical language for describing catastrophic changes in nature and technology: loss of stability of a structure when the critical load is exceeded, transition from laminar flow to turbulent, sudden collapse of a biological population when the catch quota is exceeded, trans...

For $\mu > 0$: two equilibria $x^* = \pm\sqrt{\mu}$. At $x^* = +\sqrt{\mu}$: $f'(x) = -2x|_{x = \sqrt{\mu}} = -2\sqrt{\mu} < 0$ → stable. At $x^* = -\sqrt{\mu}$: $f' = +2\sqrt{\mu} > 0$ → unstable.

For $\mu = 0$: one equilibrium $x^* = 0$, semi-stable (neutral). Bifurcation point.

Formulas

The direct Lyapunov method is not only a theoretical tool but also a practical means of design. In control theory, it enables the design of control algorithms that guarantee the stability of the closed-loop system "by construction." Unlike frequency-based methods (such as the Nyquist criterion), ...

Let us consider a nonlinear system: ẋ = f(x) + g(x)u, where x is the state and u is the control.

Lyapunov Control Method: We choose the desired Lyapunov function V(x) (for example, V = |x|²/2). We require V̇ < 0:

If ∇V · g ≠ 0, choose: u = −k(x) · (∇V · f + ε|∇V · g|) / (∇V · g), where k > 0, ε > 0.

In 1814, Laplace formulated the ideal of scientific determinism: a mind that knows the position and velocity of every particle in the universe could predict its future and reconstruct its past with arbitrary precision. By the 20th century it became clear that this ideal is unattainable—and not on...

Deterministic nonlinear systems, obeying precise mathematical equations, can exhibit chaotic behavior: the exponential divergence of close trajectories makes long-term prediction fundamentally impossible. This is deterministic chaos.

In 1963, Edward Lorenz studied a simplified model of convection in the atmosphere. The system consists of three ODEs:

With σ = 10, ρ = 28, β = 8/3, the system exhibits chaotic behavior. The trajectories never close, but remain bounded—they "twist" around two unstable equilibria, forming an infinitely thin fractal structure—the "Lorenz attractor" or the "Lorenz butterfly".

Formulas

Many real-world systems are subject to random disturbances that are fundamentally impossible to ignore. The Brownian motion of a molecule in a liquid is determined by random collisions with neighboring molecules. The price of a stock on the exchange undergoes random fluctuations under the influen...

Stochastic differential equations (SDEs) include randomness explicitly in the dynamics of the system. This is not a “model imprecision”—it is a recognition of the fundamental role of noise.

The Wiener process W(t) (Brownian motion) is a stochastic process that satisfies: 1. W(0) = 0, 2. W(t) is continuous in t (almost surely), 3. The increments W(t) − W(s) ~ N(0, t − s) for t > s, 4. Increments on non-overlapping intervals are independent.

Key property: W(t) is nowhere differentiable (almost surely). Its “derivative” dW/dt—“white noise”—does not exist in the usual sense. However, the stochastic integral ∫ f(t) dW(t) can be defined as the limit of sums using a special (Ito-iterative) procedure.