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Mathematics

Differential Equations

Bernoulli Differential Equations

Bessel's Differential Equations

Basic Differential Equations

Delay Differential Equations

Dynamical Systems and Phase Plane Analysis

First-Order Linear Differential Equations

Frobenius Method

Finite Difference Method for PDEs

Fourier Transform Methods for PDEs

Heat Equation

Laplace's Equation

Legendre's Differential Equations

Linear Second-Order Homogeneous Equations

Nonlinear Dynamics and Chaos

Perturbation Methods for Differential Equations

Poisson's Equation

Power Series Solutions of Differential Equations

Riccati Differential Equations

Sturm-Liouville Theory

Dynamical Systems and Phase Plane Analysis

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Phase plane analysis of competing species

Lotka-Volterra model, coexistence equilibrium, species competition or cooperation

Predator-prey models

Lotka-Volterra equations, oscillatory dynamics, predator-prey interactions

Dynamics of a damped pendulum

Damped harmonic oscillator, phase plane trajectories, energy dissipation

Existence of limit cycles

Poincaré-Bendixson Theorem, non-existence in linear systems, criteria for limit cycles existence

Stability of critical points

Lyapunov's direct method, stability, asymptotic stability, and instability

Bifurcation theory

Saddle-node, transcritical, pitchfork, and Hopf bifurcations, parameter changes leading to new behavior

Epidemiological models in phase plane

SIR model, endemic equilibria, threshold phenomena

Linear systems and their solutions

Eigenvalues and eigenvectors, types of nodes and spirals, direction fields

Conservative systems

Conservation of energy, closed orbits, Hamiltonian systems

Nonlinear systems and linearization

Hartman-Grobman theorem, Jacobian matrix, local behavior near critical points

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