Orientation DE Terminology Glossary of differential equations vocabulary — order, linearity, homogeneity, initial conditions, and more. Orientation ODE vs Data Fitting When to model from first principles vs. fit a curve — understanding the distinction. Orientation Linear ODE Explainer What makes an ODE linear — structure, superposition, and why it matters for solutions. Strategy ODE Solving Strategy Decision tree for classifying and solving any ODE — separable, linear, exact, homogeneous, and more. First Order Separable Equations Separating variables — the cleanest first-order technique when f(x) and g(y) factor apart. First Order · Applications Growth & Decay Exponential models — population, radioactive decay, compound interest, RC discharge. First Order · Applications Cooling & Mixing Newton's law of cooling and tank mixing problems as first-order linear ODEs. First Order Integrating Factors Multiplying through by μ(x) to make a non-exact equation exact — derivation and method. First Order Exact Differentials When M dx + N dy = 0 is exact, and how to find the potential function F(x,y). First Order · Example Exact Differentials — Example Full worked example solving an exact differential equation from identification to solution. First Order · Connections Gradient, Conservative & Exact How exact equations, conservative vector fields, and gradient fields are all the same idea. First Order · Numerical Euler's Method Step-by-step numerical integration — building solutions one slope at a time. First Order · Theory Existence & Uniqueness — Nonlinear When are solutions guaranteed? Picard's theorem and what breaks down for nonlinear equations. Second Order · Foundations First & Second Order — Deep Dive Foundational first-principles treatment connecting first and second-order ODE theory. Second Order Second-Order ODEs Complete treatment of 2nd-order linear ODEs — structure, solution space, and all root cases. Second Order · Foundation Euler's Formula e^(iθ) = cos θ + i sin θ — derivation, geometric meaning, and how it powers complex root solutions. Second Order Constant Coefficient Homogeneous The characteristic equation — real distinct, repeated, and complex roots, all three cases. Second Order · Example Distinct Real Roots — Step by Step Full worked solution for the distinct real roots case from characteristic equation to general solution. Second Order · Proofs ODE Roots — Proofs Rigorous derivations showing why the three root cases produce the solution forms they do. Second Order · Principle Superposition Why linear ODEs allow solution sums — the superposition principle derived from linearity. Nonhomogeneous Nonhomogeneous ODEs — Unified y = y_h + y_p — the structure of nonhomogeneous solutions and how the pieces fit together. Advanced Technique Reduction of Order Given one solution y₁, find y₂ by substituting y = v(x)y₁ — reducing order by one. Advanced Technique Wronskian & Cramer's Rule The Wronskian as a determinant of independence — connecting ODEs to linear algebra via Cramer's rule. Advanced Technique Variation of Parameters Finding y_p when undetermined coefficients fails — the general method using Wronskian integrals. Advanced Technique Cauchy–Euler Equations The variable-coefficient ODE ax²y″ + bxy′ + cy = 0 — substitution x = eᵗ and the characteristic equation. Reference Homogeneous Linear ODEs Homogeneous linear ODE structure and Wronskian — consolidated reference. Reference ODE Unified Consolidated view connecting first-order, second-order, and systems of ODEs. Systems Homogeneous Systems Systems of first-order linear ODEs — eigenvalue method and phase-plane geometry. Systems Linear Systems — Intuition Geometric and physical intuition for linear ODE systems — what eigenvalues tell you about trajectories.