Differential Equations on steroids

Module 1: The Language of Change (Foundations)

Structure & Classification

Definitions

• Ordinary Differential Equations (ODE) vs Partial Differential Equations
• Order of an ODE ($F(x,y,y^{\prime },\dots ,y^{(n)})=0$)

The “Grammar” of ODEs

• Explicit vs Implicit: $y^{(n)}=G(...)$ vs $F(...)=0$
• Linearity: Linear vs Nonlinear
• Time Dependence: Autonomous ($y^{\prime }=f(y)$) vs Non-autonomous ($y^{\prime }=f(t,y)$)
• Homogeneity: Homogeneous ($g(x)=0$) vs Inhomogeneous ($g(x)\ne 0$)

The Geometric View

Slope Fields (Direction Fields)

• Visualizing $y^{\prime }=F(x,y)$ as a vector field
• Isoclines (lines of constant slope)
• Key Insight: Autonomous equations have slope fields invariant under x-translation

Existence & Uniqueness

Initial Value Problems (IVP)

• Formulation: ODE + Initial Condition $y(x_0)=y_0$

Picard-Lindelöf Theorem

• Requirements: Continuity + Lipschitz condition
• Consequence: Solution trajectories cannot cross in the phase space
• Dependence on initial conditions (Gronwall’s inequality concept)

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Module 2: First-Order Dynamics (1D Systems)

Analytical Solvers (The Toolbox)

Separable Equations

• Form: $y^{\prime }=g(x)h(y)$
• Method: $\int \frac{1}{h(y)}dy=\int g(x)dx$
• Application: Logistic Growth $\dot{N}=kN(A-N)$

Linear Equations (First Order)

• Form: $y^{\prime }+f(x)y=g(x)$
• Method 1: Variation of Constants
• Method 2: Integrating Factor $\mu (x)=e^{\int f(x)dx}$

Exact Equations

• Form: $M(x,y)+N(x,y)y^{\prime }=0$
• Condition: $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ (Potential function $\psi$)
• Technique: Integrating factors for non-exact equations

Special Substitutions

• Bernoulli Equations: $y^{\prime }+g(x)y+h(x)y^{\alpha }=0$
• Riccati Equations: $y^{\prime }+g(x)y+h(x)y^2=k(x)$
• Homogeneous type: $y^{\prime }=F(y/x)$ using $u=y/x$

Discrete Time Systems (1D)

Difference Equations

• Form: $x_{n+1}=f(x_n)$
• Cobweb Diagrams: Graphical analysis of iteration
• Application: Discrete Logistic Map $x_{n+1}=r{x}_n(1-x_n)$

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Module 3: Linear Theory & Oscillations (2nd Order)

Structure of Linear Solutions

Superposition Principle

• General Solution = Homogeneous Solution ($y_h$) + Particular Solution ($y_p$)
• Vector space structure of $y_h$ (Dimension = Order $n$)

Homogeneous Linear Equations (Constant Coefficients)

The Characteristic Polynomial

• $P(\lambda )={\lambda }^n+a_{n-1}{\lambda }^{n-1}+\cdots +a_0=0$

Solution Cases

• Real Distinct Roots: $y=C_1{e}^{{\lambda }_{1}t}+C_2{e}^{{\lambda }_{2}t}$
• Repeated Roots: $y=C_1{e}^{\lambda t}+C_{2}t{e}^{\lambda t}$
• Complex Roots: $y=e^{\alpha t}(C_1\cos (\beta t)+C_2\sin (\beta t))$ where $\lambda =\alpha \pm i\beta$

Inhomogeneous Linear Equations (Forced Systems)

Method of Undetermined Coefficients (Ansatz)

• Polynomials $\to$ Polynomials
• Exponentials $\to$ Exponentials (watch for Resonance!)
• Sin/Cos $\to$ $A\sin (\omega t)+B\cos (\omega t)$

Variation of Constants

• General formula using the Wronskian logic

Laplace Transforms

• Transfer function $Y(s)=G(s)U(s)$

Oscillation Problems (Physics)

Free Oscillations (Homogeneous)

• Undamped: Harmonic motion (${\omega }_0=\sqrt{k/m}$)
• Damped: Overdamped vs Underdamped vs Critically Damped

Forced Oscillations (Inhomogeneous)

• Amplitude response & Resonance catastrophe
• Beat effects (near-resonance)

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Module 4: Systems & Phase Plane Geometry (2D)

Fundamentals of Systems

Transformation

• Reducing high-order ODEs to 1st-order systems: $y_1=y,y_2=y^{\prime }$

Vector Fields

• Phase Portraits for Autonomous systems $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$

Linear Systems Analysis

Matrix Formulation

• $\dot{\mathbf{x}}=A\mathbf{x}$

Eigenvalues and Stability

• Real Eigenvalues: Nodes (Stable/Unstable) and Saddle Points
• Complex Eigenvalues: Spirals (Stable/Unstable) and Centers
• Repeated Eigenvalues: Degenerate Nodes / Stars

Analytical Solution

• Matrix Exponential: $\mathbf{x}(t)=e^{At}{\mathbf{x}}_0$
• Decoupling via diagonalization

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Module 5: Nonlinear Dynamics & Chaos

Linearization (Local Stability)

The Jacobian Matrix

• $J=\left(\begin{array}{cc}\partial f_1/\partial x_1& \partial f_1/\partial x_2\\ \partial f_2/\partial x_1& \partial f_2/\partial x_2\end{array}\right)$ evaluated at fixed point ${\mathbf{x}}^{\ast }$

Hartman-Grobman Theorem

• Linearization predicts behavior for Hyperbolic fixed points (Re($\lambda$) $\ne 0$)
• Trace-Determinant Plane classification

Global Behavior (2D)

Conservative Systems

• Energy functions $E(x,y)$ = const
• Fixed points are centers or saddles (never nodes/spirals)

Limit Cycles

• Isolated closed orbits (Van der Pol Oscillator)
• Poincaré-Bendixson Theorem: Trapping regions imply closed orbits

Bifurcations

1D Bifurcations

• Saddle-Node (Creation/Destruction of fixed points)
• Transcritical (Stability exchange)
• Pitchfork (Symmetry breaking: Supercritical vs Subcritical)

2D Bifurcations

• Hopf Bifurcation: Birth of a limit cycle from a spiral point

Chaos (3D and Discrete)

Continuous Chaos (3D+)

• Lorenz Attractor (Butterfly effect)
• Rössler System
• Strange Attractors and fractal geometry

Discrete Chaos (1D Maps)

• Logistic Map: Period doubling cascade
• Feigenbaum Constant $\delta \approx 4.669$
• Lyapunov Exponents (measuring sensitivity to initial conditions)
• Complex Dynamics: Mandelbrot & Julia Sets

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Module 6: Numerical Methods

One-Step Methods

Explicit Euler

• $x_{k+1}=x_k+hf(t_k,x_k)$
• Error Order: $O(h)$ (Global order 1)

Heun’s Method (Improved Euler)

• Predictor-Corrector approach
• Error Order: $O(h^2)$

Runge-Kutta (RK4)

• Standard 4-stage method (The workhorse of ODEs)
• Error Order: $O(h^4)$

Stability & Accuracy

Error Analysis

• Local Discretization Error (${\tau }_{k,h}$) vs Global Error ($e_k$)
• Consistency: Local error $\to 0$ as $h\to 0$

Stiff Systems

• Problem: Multiple timescales (fast/slow dynamics) causing instability in explicit methods
• Solution: Implicit Methods (e.g., Implicit Euler)
  • Region of Absolute Stability
  • Step size control (Adaptive methods like ode45)