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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

Finite Difference Method for PDEs

Frobenius Method

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

Power Series Solutions of Differential Equations

Poisson's Equation

Riccati Differential Equations

Sturm-Liouville Theory

Dynamical Systems and Phase Plane Analysis

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Linear systems and their solutions

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

Predator-prey models

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

Dynamics of a damped pendulum

Damped harmonic oscillator, phase plane trajectories, energy dissipation

Stability of critical points

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

Existence of limit cycles

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

Epidemiological models in phase plane

SIR model, endemic equilibria, threshold phenomena

Nonlinear systems and linearization

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

Conservative systems

Conservation of energy, closed orbits, Hamiltonian systems

Phase plane analysis of competing species

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

Bifurcation theory

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

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