Differential equations: Introduction

Differential equations

Differential equations versus algebraic equations. Famous Examples of Differential Equations. Classification of Differential Equations. Solving First-Order Ordinary Differential Equations. Read more... 1938 words,🕔10 minutes read, May 14, 2022.

Separable differential equations

Solving differential equations. Simple Harmonic Motion, A Pendulum (EDO Intuition). Separation of Variables. Steps to Solve Separable Differential Equations. Solved exercises. Read more... 1989 words,🕔10 minutes read, May 14, 2022.

Separable Differential Equations: Solved Exercises and Applications

Differential Equations, Separable Differential Equations, Newton's Law of Cooling, Orthogonal Trajectories, Calculus, Separation of Variables, Exponential Decay, Slope Fields, Thermodynamics, Initial Value Problems, Tangent Lines, Geometric Modeling, Integration, First-Order ODEs Read more... 2156 words,🕔11 minutes read, May 14, 2022.

Bernoulli Differential Equations: Methods and Solved Exercises

Detailed guide on solving Bernoulli differential equations, a specific type of nonlinear first-order ODE. Covers the standard form, the substitution method to linearize the equation, and the homogeneous solution method with step-by-step derivations and solved exercises. Read more... 2336 words,🕔11 minutes read, May 14, 2022.

Integrating Factors for Linear Differential Equations

A comprehensive guide to solving first-order linear differential equations using the Integrating Factor method. Includes the step-by-step derivation, solved exercises, and an application to Newton's Law of Cooling Read more... 2131 words,🕔11 minutes read, May 14, 2022.

Exact Differential Equations and Integrating Factors

Comprehensive guide to solving exact differential equations. Explains the exactness test, potential functions, and techniques for transforming non-exact ODEs into exact equations using integrating factors (dependent on x, y, or xy), complete with solved exercises. Read more... 2414 words,🕔12 minutes read, May 14, 2022.

Geometric Interpretation of Differential Equations

A geometric approach to First-Order Ordinary Differential Equations (ODEs). Covers direction fields, isoclines, and integral curves to visualize solution behaviors, including circles and asymptotic stability without solving explicitly. Read more... 3043 words,🕔15 minutes read, May 17, 2022.

Existence and Uniqueness Theorem

Explores the Picard–Lindelöf Existence and Uniqueness Theorem for first-order ODEs. Features a comparison between a linear equation solved via integrating factors and a separable equation demonstrating singularities and failure of uniqueness. Read more... 1812 words,🕔9 minutes read, May 17, 2022.

Geometric Interpretation of Differential Ordinary Equations 2

Explores the geometric interpretation and analytical solutions of first-order Ordinary Differential Equations (ODEs). Analyzes direction fields and isoclines to understand asymptotic behavior for linear equations like y' = x+y and nonlinear equations like y' = -y/(x²+y²). Covers the concept of the separatrix, singularities, and solving Initial Value Problems using integrating factors. Read more... 3180 words,🕔15 minutes read, May 17, 2022.

Improved Euler's Method or RK2

Improved Euler's Method (Heun's Method or RK2) for approximating ODEs. Step-by-step examples, cost vs. benefit of accuracy, pitfalls with singularities, and implications for numerical methods. Read more... 2223 words,🕔11 minutes read, May 17, 2022.

First-Order Linear ODEs and Applications

First-Order Linear Ordinary Differential Equations (ODEs). Standard form, existence and uniqueness, and solving via integrating factors. Applications including Newton's Law of Cooling and the Diffusion Model for salt concentration. Read more... 2700 words,🕔13 minutes read, May 17, 2022.

Solving First-Order Linear ODEs. Integrating Factors

Step-by-step guide to solving first-order linear ordinary differential equations (ODEs) using the method of integrating factors. Includes worked examples and initial value problems. Read more... 1162 words,🕔6 minutes read, May 17, 2022.

First-order Substitution Methods

Solving nonlinear first-order differential equations using direct and inverse substitution methods. Step-by-step examples transforming equations into separable forms, integrating using partial fractions, and identifying singular solutions. Read more... 2034 words,🕔10 minutes read, May 17, 2022.

First-Order Substitution Methods. Rescaling and Bernoulli Equations

Solving first-order ODEs using substitution methods. Covers rescaling a nonlinear temperature model (Stefan-Boltzmann law) with dimensionless variables and solving Bernoulli differential equations via direct substitution. Read more... 2199 words,🕔11 minutes read, May 17, 2022.

Homogeneous First-Order ODE's

Explore homogeneous first-order ODEs, their scale invariance, and solving techniques using y/x substitution. Includes solved exercises and a real-world application modeling the path of a boat under a lighthouse beam as an exponential spiral. Read more... 2336 words,🕔11 minutes read, May 17, 2022.

First-Order Autonomous ODEs II. The Logistic Equation & Harvesting

Explore the Logistic Differential Equation and its application in modeling population growth with carrying capacity. Includes qualitative analysis, explicit solutions, and an in-depth analysis of the logistic equation with constant harvesting (sustainable, critical threshold, and over-harvesting). Read more... 2947 words,🕔14 minutes read, May 17, 2022.

First-order Linear with Constant Coefficients

Solving first-order linear ODEs with constant coefficients using integrating factors. Distinguishes between particular (steady-state) and homogeneous (transient) responses, and demonstrates the Superposition Principle with step-by-step examples. Read more... 2617 words,🕔13 minutes read, May 17, 2022.

Differential Equations with Periodic Inputs

Solving first-order linear differential equations with periodic inputs using complexification and Euler's formula. Analyze steady-state solutions, amplitude attenuation, and phase lag through polar and Cartesian forms, with proofs for the trigonometric identity. Read more... 2570 words,🕔13 minutes read, May 17, 2022.

Second-order Linear ODE's with Constant Coefficients

Solving second-order linear homogeneous ordinary differential equations with constant coefficients. Covers the characteristic equation method, real and distinct roots (overdamped systems), complex roots (underdamped systems), and the theorem on real and imaginary parts of complex solutions. Includes physical interpretations using the mass-spring-damper model. Read more... 2623 words,🕔13 minutes read, May 17, 2022.

Complex Numbers and Differential Equations

A comprehensive guide to complex numbers, covering algebraic operations, polar form, Euler's formula, laws of exponents, and the differentiation and integration of complex-valued functions. Includes solving first-order differential equations with complex coefficients and interpreting exponential expressions with complex exponents. Read more... 2325 words,🕔11 minutes read, May 17, 2022.

Complex Numbers and Differential Equations 2

Learn how to integrate real functions using complex exponentials to bypass tedious integration by parts. Explore the N-th roots of unity, their geometric interpretation as regular polygons on the unit circle, primitive roots, and key algebraic properties. Read more... 2292 words,🕔11 minutes read, May 17, 2022.

Basic Linear ODE. Mixing Models and Applications

Explore basic linear ODEs, their common forms, and real-world applications. Learn how to model mixing problems like salt in a tank or alcohol in a vat using the conservation of mass, and solve for concentration dynamics under constant input. Read more... 2737 words,🕔13 minutes read, May 17, 2022.

Second-order Linear ODE's. Two equal roots (Critically damped system).

Solving second-order linear homogeneous ODEs with constant coefficients when the characteristic equation has two equal roots (critically damped system). Covers finding the second independent solution using the Reduction of Order Method and Direct Substitution, and explores the physical interpretation of critical damping. Read more... 2114 words,🕔10 minutes read, May 17, 2022.

Basic linear ODE. Electrical Circuits (RC Circuit).

Analysis of first-order linear ODEs applied to electrical RC circuits. Derives the governing equation using Kirchhoff's Voltage Law, solves for a constant voltage source, and interprets the transient and steady-state response, including the time constant and a hydraulic metaphor. Read more... 2131 words,🕔11 minutes read, May 17, 2022.

Second-order Linear Homogeneous ODE's. Complex Roots and Damped Oscillations

Exploring second-order linear homogeneous ODEs with constant coefficients, focusing on complex characteristic roots and alternative complex constants. Discusses the physical significance of underdamped systems, and derives the equations for undamped and damped oscillations using the mass-spring model. Read more... 2202 words,🕔11 minutes read, May 17, 2022.

Radioactive Decay Chains and Modeling

Modeling radioactive decay chains (A→B→C) using first-order linear ordinary differential equations. Solving for substance dynamics with the integrating factor method, analyzing transient and secular equilibrium regimes, and exploring growing systems with constant source terms. Read more... 1856 words,🕔9 minutes read, May 17, 2022.

Solving the Damped Oscillator Equation. Cases Based on Damping

Solving the damped mass-spring oscillator equation and analyzing the different cases based on damping, namely overdamped, critically damped, underdamped, and undamped (simple harmonic motion) systems Read more... 1487 words,🕔7 minutes read, May 17, 2022.

Solving First-Order Linear Ordinary Differential Equations with Sinusoidal Inputs

Solving first-order linear ordinary differential equations with sinusoidal inputs using the method of complex exponentials and integrating factors. Analyzing steady-state amplitude and phase lag, and applying the superposition principle for combined inputs. Read more... 2087 words,🕔10 minutes read, May 17, 2022.

Euler's Method. Numerical Approximation for ODEs

Learn Euler's Method for approximating solutions to first-order ODEs and Initial Value Problems (IVPs). Includes step-by-step examples, error analysis, convexity checks, and ways to improve numerical accuracy. Read more... 2215 words,🕔11 minutes read, May 17, 2022.

General Solution of a Second-Order Linear Homogeneous ODE

Constructing the general solution of a second-order linear homogeneous ODE. Covers linear independence, the Wronskian test, the superposition principle (with two formal proofs using direct substitution and linear operators), the existence and uniqueness theorem, and worked examples for both constant and variable coefficient (Cauchy-Euler) equations. Read more... 2801 words,🕔14 minutes read, May 17, 2022.

General Solution, Uniqueness, and Normalized Solutions of Second-Order ODEs

Deep dive into the general solution of second-order linear homogeneous ODEs. Covers solving the initial value problem using the Wronskian, Abel's Identity, the completeness of the solution set, and the computational power of normalized solutions with oscillatory (trig) and saddle (hyperbolic) examples, grounded in the Existence and Uniqueness Theorem. Read more... 2546 words,🕔12 minutes read, May 17, 2022.

Second-Order Linear Differential Equations. Homogeneous vs. Inhomogeneous

Learn how to solve second-order linear inhomogeneous differential equations by combining complementary and particular solutions. Explore real-world applications including mass-spring-dashpot systems and RLC circuits, along with the method of undetermined coefficients and worked examples. Read more... 2926 words,🕔14 minutes read, May 17, 2022.

Solving First-Order Linear Inhomogeneous ODEs and Stability Analysis

Learn how to solve first-order linear inhomogeneous ODEs using the method of integrating factors. Covers complementary and particular solutions, analysis of long-term behavior, and stability (stable vs. unstable systems), with detailed worked examples. Read more... 1663 words,🕔8 minutes read, May 17, 2022.

Partial differential equations

Classification of differential equations. Ordinary Differential Equations. The heat and the harmonic oscillator equations. First-order differential equations. Read more... 3003 words,🕔15 minutes read, May 14, 2022.

Stability Criteria for Second-Order Linear Differential Equations

Analyze the stability of solutions to second-order linear differential equations with constant coefficients. Covers how characteristic roots (distinct, repeated, complex) dictate system behavior, definitions of stable, unstable, and marginally stable systems, and physical interpretations using spring-mass-dashpot and RLC circuit examples. Read more... 2033 words,🕔10 minutes read, May 17, 2022.

First-order Autonomous ODE's

First-order Autonomous ODEs. Definition, qualitative analysis, phase lines, and equilibrium points (stable, unstable). Includes solved examples like dy/dt = 1 - y^2 and a real-world bank account embezzlement model. Read more... 3426 words,🕔17 minutes read, May 17, 2022.

Particular Solutions Inhomogeneous ODEs. Exponential Input and Resonance

Learn how to find particular solutions for second-order linear inhomogeneous ODEs with constant coefficients. Covers differential operator notation, the Substitution Rule, the Exponential Input Theorem, handling complex forcing terms, and the Exponential Shift Rule for solving resonance cases. Read more... 2116 words,🕔10 minutes read, May 17, 2022.

Resonance in Differential Equations

Explore the phenomenon of resonance in second-order linear inhomogeneous differential equations. Covers finding particular solutions for non-resonant and resonant cases using complex exponentials, deriving the resonant solution via L'Hôpital's Rule, and understanding the geometric interpretation of beats. Read more... 1853 words,🕔9 minutes read, May 17, 2022.

Complex Eigenvalues in Systems of Differential Equations

Complex Eigenvalues in Systems of Differential Equations. Examples. Modelling a love affair. Read more... 1831 words,🕔9 minutes read, May 17, 2022.

Introduction to Fourier Series and Coefficients

Introduction to Fourier series for representing periodic functions as infinite sums of sines and cosines. Covers the superposition principle for solving linear ODEs, the orthogonality of trigonometric functions, and the derivation of Euler's formulas for Fourier coefficients Read more... 1959 words,🕔10 minutes read, May 17, 2022.

Fourier Series. Square-Wave, Gibbs Phenomenon, & Uniqueness

Explore the Fourier series of the square-wave function, understand discontinuities and the Gibbs phenomenon, and learn about the uniqueness of Fourier series and the mathematical conditions required for their existence. Read more... 1907 words,🕔9 minutes read, May 17, 2022.

Predator-Prey Model (Lotka-Volterra)

Detailed analysis of the Lotka-Volterra predator-prey model. Covers finding and linearizing critical points, Trace-Determinant analysis revealing centers and structural instability, eliminating time to find conserved quantities and closed trajectories, and applying Volterra's Principle to explain the paradox of intervention in ecological systems like fishing and pesticide use. Read more... 2581 words,🕔13 minutes read, May 17, 2022.

Fourier Series. Symmetry, Convergence, and Arbitrary Periods

Explore Fourier series for even and odd functions, symmetry shortcuts for coefficients, Dirichlet's conditions for convergence, the Gibbs phenomenon at discontinuities, and how to extend Fourier series to arbitrary periods (2L) using half-range expansions. Read more... 2124 words,🕔10 minutes read, May 17, 2022.

Fourier Series. Even/Odd Extensions & Examples.

Even and Odd Extensions of Functions. Fourier Sine and Cosine Series. Sawtooth Wave and Rectangular Pulse. Gibbs Phenomenon and Convergence of Fourier Series Read more... 1682 words,🕔8 minutes read, May 17, 2022.

Finding Particular Solutions via Fourier Series

Finding Particular Solutions via Fourier Series. Undamped harmonic oscillator driven by a square wave forcing function. Principle of Superposition and Resonance phenomenon. Direct substitution method. Read more... 2016 words,🕔10 minutes read, May 17, 2022.

Damped harmonic oscillator

Damped harmonic oscillator and the three damping regimes (underdamped, critically damped, overdamped). Finding the general solution to y'' + 3y = 2x using Fourier series. Read more... 2106 words,🕔10 minutes read, May 17, 2022.

Introduction to the Laplace Transform

Introduction to Fourier Series and Euler's formulas. Definition of the Laplace Transform from power series, linearity, exponential shift formula, and transforms of unit step, window, and Dirac delta functions. Second shifting theorem and time delays. Read more... 2075 words,🕔10 minutes read, May 17, 2022.

Advanced and Inverse Laplace Transforms

Advanced Laplace Transforms including complex exponentials and trigonometric functions. Key operational properties like time scaling and frequency integration. Methods for finding the Inverse Laplace Transform. Partial Fractions, Exponential Shift, and Completing the Square. Read more... 1704 words,🕔8 minutes read, May 17, 2022.

Laplace Transforms of Polynomials. Existence and Exponential Order.

Laplace Transforms of Polynomials using recursive integration by parts and the Gamma Function. Conditions for the existence of the Laplace Transform, including piecewise continuity and exponential order. Examples and counterexamples of functions of exponential type. Read more... 1832 words,🕔9 minutes read, May 17, 2022.

Solving Linear ODE's using the Laplace Theorem

Solving linear ordinary differential equations (ODEs) and Initial Value Problems (IVPs) using the Laplace Transform. Laplace Transform of first and higher-order derivatives. Step-by-step solution of y'' - y = e^(-t) using partial fraction decomposition. Read more... 1859 words,🕔9 minutes read, May 17, 2022.

Solving Linear ODEs using the Laplace Transform. Worked Examples

Step-by-step solutions for linear ordinary differential equations using the Laplace Transform. Solved exercises including second-order ODEs with real and complex roots, and a fourth-order ODE, utilizing partial fraction decomposition. Read more... 1205 words,🕔6 minutes read, May 17, 2022.

Convolution. Definition, Theorem, and Intuition

Definition and basic properties of convolution. Proof of the Convolution Theorem linking convolution to the Laplace Transform. Solved mathematical examples and intuitive real-world applications like radioactive decay and cumulative growth. Read more... 2275 words,🕔11 minutes read, May 17, 2022.

Heaviside Step Function and Time-Shifting Theorem

Jump discontinuities and the Heaviside step function. Laplace Transform of the unit step and unit box functions. The T-axis translation formula (Time-Shifting Theorem) with proof and solved exercises. Read more... 2339 words,🕔11 minutes read, May 17, 2022.

Convolution. Impulse Inputs and the Dirac Delta Function

Properties of convolution, modeling impulse inputs with box functions, and the definition of the Dirac Delta function. Covers the sifting property, Laplace transform of the delta function, and convolution with the delta function as the identity element. Read more... 2165 words,🕔11 minutes read, May 17, 2022.

Solving ODEs with Impulses and Transfer Function

Solving undamped mass-spring systems and ODEs with periodic impulses using Laplace Transforms. Introduction to the general second-order linear differential equation, transfer functions, weight functions, and convolution in the time domain. Read more... 1948 words,🕔10 minutes read, May 17, 2022.

Linear and Autonomous First-Order Systems of ODEs

An in-depth guide to first-order systems of ODEs, covering linear systems, the elimination and operator methods for solving them, autonomous systems, equilibrium points, and geometric interpretations in the phase plane. Read more... 2902 words,🕔14 minutes read, May 17, 2022.

Homogeneous Linear Systems with Constant Coefficients

An in-depth guide to solving homogeneous linear systems of ODEs with constant coefficients using the matrix approach. Covers formulating the eigenvalue problem, the trace-determinant shortcut, finding eigenvectors, and analyzing dynamical behavior like saddle points. Read more... 2673 words,🕔13 minutes read, May 17, 2022.

Homogeneous Linear Systems with repeated real eigenvalues

A guide to solving homogeneous linear systems of ODEs with repeated real eigenvalues. Covers algebraic and geometric multiplicity, complete vs. defective eigenvalues, and the Spectral or Principal Axis Theorem for symmetric matrices, featuring a 3x3 fish tank heat exchange model. Read more... 1873 words,🕔9 minutes read, May 17, 2022.

Sketching Solutions of 2x2 Homogeneous Linear Systems

A guide to sketching phase portraits for 2x2 homogeneous linear systems of ODEs with constant coefficients. Covers finding eigenvalues and eigenvectors, analyzing stability, classifying equilibrium points (nodes, sinks, sources), and understanding dominant eigenvalue behavior using a tourism competition model. Read more... 2486 words,🕔12 minutes read, May 17, 2022.

Sketching Solutions of Homogeneous Linear Systems. Complex Eigenvalues

A guide to sketching phase portraits for 2x2 homogeneous linear systems with complex eigenvalues. Covers deriving general solutions using Euler's formula, determining rotation direction, and analyzing stability to classify equilibrium points as spiral sinks or spiral sources. Read more... 1586 words,🕔8 minutes read, May 17, 2022.

Sketching Solutions of Homogeneous Linear System 2

A continuation of sketching phase portraits for 2x2 homogeneous linear systems. Covers classifying equilibrium points as stable nodes, unstable nodes, and saddle points by analyzing eigenvalues, eigenvectors, and dominant terms, with real-world applications in tourism competition and advertising wars. Read more... 2077 words,🕔10 minutes read, May 17, 2022.

General Homogeneous Linear Systems. Wronskian, Fundamental Matrix, and Centers.

An exploration of general homogeneous linear systems of ODEs, covering the Principle of Superposition, linear independence, the Wronskian determinant, and the fundamental matrix. Includes an example of sketching phase portraits for systems with purely imaginary eigenvalues, resulting in closed orbits (centers). Read more... 1430 words,🕔7 minutes read, May 17, 2022.

Solving Inhomogeneous Systems of ODEs

A guide to solving inhomogeneous systems of linear differential equations with constant coefficients. Covers matrix representation, the general solution structure (complementary and particular), Variation of Parameters using the fundamental matrix, and the Method of Undetermined Coefficients, illustrated with a two-tank mixing problem. Read more... 2772 words,🕔14 minutes read, May 17, 2022.

Matrix Exponentials and Fundamental Matrices

An in-depth guide to solving linear systems of ODEs using matrix exponentials. Covers the fundamental matrix solution, the power series definition, diagonalization, the exponential law for matrices, and practical methods for computing e^At. Read more... 2811 words,🕔14 minutes read, May 17, 2022.

Sketching Non-linear Autonomous Systems

An exploration of sketching non-linear autonomous systems of differential equations using a lightly damped pendulum as an example. Covers finding critical points, linearizing systems with the Jacobian matrix, calculating eigenvalues, and classifying local behaviors like spiral sinks and saddle points. Read more... 1830 words,🕔9 minutes read, May 17, 2022.

Non-linear Autonomous Systems. Predator-Prey Analysi

An in-depth analysis of a non-linear predator-prey autonomous system. Covers finding critical points, linearizing the system using the Jacobian matrix, calculating eigenvalues to determine stability (unstable nodes, saddle points, stable spirals), and interpreting the ecological meaning of the mathematical results. Read more... 1984 words,🕔10 minutes read, May 17, 2022.

Limit Cycles in Non-Linear Autonomous Systems

Limit Cycles in Non-Linear Autonomous Systems. Understanding the phase plane, velocity fields, and trajectories. Analysis of critical points and local stability via the Jacobian matrix. Distinction between closed trajectories and isolated limit cycles, with examples including the simple harmonic oscillator. Read more... 2724 words,🕔13 minutes read, May 17, 2022.

Bendixson's Criterion and Limit Cycle Non-Existence

Bendixson's Criterion and the Critical Point Criterion for proving the non-existence of limit cycles in non-linear autonomous systems. Analysis of vector field divergence, proof via Green's Theorem, and solved exercises including spring-mass-damper systems with non-linear damping. Read more... 2173 words,🕔11 minutes read, May 17, 2022.

Relation Between Non-linear Autonomous Systems and First-order ODEs

Converting non-linear autonomous systems into first-order ODEs by eliminating the time parameter. Qualitative analysis including critical point identification, linearization via the Jacobian matrix, and eigenvalue classification. Application to the simple harmonic oscillator yielding circular trajectories. Read more... 1477 words,🕔7 minutes read, May 17, 2022.

The Trace-Determinant Plane

Classifying linear systems of differential equations using the Trace-Determinant plane. Analyzing eigenvalues, the discriminant, and equilibrium points (nodes, spirals, saddles, centers). Includes similarity transformations, canonical forms, and a detailed solution for systems with complex eigenvalues. Read more... 2642 words,🕔13 minutes read, May 17, 2022.

The Trace-Determinant Plane. Equilibrium Point Classification

Solving 2x2 linear systems and classifying equilibrium points based on eigenvalues. Detailed analysis of distinct real eigenvalues (nodes, saddle points), repeated real eigenvalues (star nodes, defective/improper nodes), and complex roots (spirals, centers) using the Trace-Determinant plane. Read more... 1715 words,🕔9 minutes read, May 17, 2022.

Decoupling Linear Systems with Constant Coefficients

A guide to decoupling linear systems of ordinary differential equations (ODEs) with constant coefficients using matrix diagonalization. Covers changing variables, finding eigenvalues and eigenvectors, and interpreting steady-state and transient solutions with physical examples like the ice cube tray problem. Read more... 2506 words,🕔12 minutes read, May 17, 2022.

Decoupling Linear Systems with Constant Coefficients 2

A step-by-step worked example of decoupling a linear system of ODEs with constant coefficients using matrix diagonalization. Covers finding eigenvalues and eigenvectors, applying a change of variables, solving the independent equations, and analyzing the system's behavior as an unstable node. Read more... 1191 words,🕔6 minutes read, May 17, 2022.