Linear differential equation Study Guide

📖 Core Concepts Linear differential equation – an equation linear in the unknown function \(y\) and its derivatives: \[ a0(x)y + a1(x)y' + \dots + an(x)y^{(n)} = b(x) \] Order – highest derivative appearing (e.g., second‑order if \(y''\) is present). Homogeneous vs. non‑homogeneous – homogeneous: \(b(x)\equiv0\); non‑homogeneous: \(b(x)\neq0\). Linear differential operator – \(L = a0(x)+a1(x)D+\dots +an(x)D^n\) with \(D=\frac{d}{dx}\). Kernel of \(L\) – set of solutions to \(Ly=0\); forms an \(n\)-dimensional vector space (dimension = order). General solution – sum of a particular solution \(yp\) and the homogeneous solution (linear combination of basis functions). 📌 Must Remember Characteristic polynomial for constant‑coefficient homogeneous ODE: \[ a0 + a1\alpha + a2\alpha^2 + \dots + an\alpha^n = 0 \] Distinct roots → basis \(\{e^{\alphai x}\}\). Repeated root \(\alpha\) of multiplicity \(m\) → \(\{e^{\alpha x}, xe^{\alpha x},\dots ,x^{m-1}e^{\alpha x}\}\). Complex conjugate pair \(a\pm ib\) → real basis \(e^{ax}\cos(bx),\; e^{ax}\sin(bx)\). Second‑order discriminant \(D=a^2-4b\): \(D>0\): two real, distinct roots. \(D=0\): repeated real root → \(e^{\alpha x}, xe^{\alpha x}\). \(D<0\): complex pair → \(e^{ax}\cos(bx), e^{ax}\sin(bx)\). Integrating factor for first‑order linear ODE \(y'+f(x)y=g(x)\): \[ \mu(x)=e^{\int f(x)\,dx} \] Fundamental matrix for constant (or commuting) coefficient system: \(U(x)=e^{\int A(x)\,dx}\). Cauchy–Euler substitution: \(x=e^{t}\) (\(t=\ln x\)) converts the Euler equation to constant‑coefficient form. 🔄 Key Processes Solve homogeneous constant‑coeff ODE Write characteristic polynomial. Find roots \(\alphai\). Build basis using rules for distinct, repeated, and complex roots. Form general homogeneous solution \(yh=\sum ci\phii(x)\). Find particular solution (non‑homogeneous) Identify right‑hand side \(f(x)\). Choose method: Undetermined coefficients → guess form matching \(x^m e^{\beta x}, x^m\cos(\beta x), x^m\sin(\beta x)\). Annihilator → apply operator that annihilates \(f(x)\), solve extended homogeneous equation, then reduce. Variation of parameters → compute \(yp = \sum yi(x)\int \frac{Wi(x)}{W(x)}f(x)\,dx\) (where \(W\) is the Wronskian). Add to homogeneous solution: \(y = yh + yp\). First‑order linear ODE Compute integrating factor \(\mu(x)=e^{\int f(x)dx}\). Multiply: \(\frac{d}{dx}[\mu y]=\mu g(x)\). Integrate: \(y = \mu^{-1}\!\bigl(C+\int \mu g\,dx\bigr)\). Linear system \(Y'=A(x)Y+B(x)\) Solve homogeneous part: find fundamental matrix \(U(x)\). Particular solution: \(Yp = U(x)\int U^{-1}(x)B(x)\,dx\). General solution: \(Y = U(x)C + Yp\). Cauchy–Euler equation Substitute \(x=e^{t}\) (\(t=\ln x\)). Transform to constant‑coefficient ODE in \(t\). Solve, then revert: \(y(x)=\tilde y(\ln x)\). 🔍 Key Comparisons Undetermined coefficients vs. Variation of parameters Undetermined coefficients: quick, works only when \(f(x)\) is a linear combination of exponentials, polynomials, sines/cosines. Variation of parameters: always works; requires knowledge of homogeneous basis and integration of Wronskian terms. Distinct vs. repeated roots Distinct: one exponential per root. Repeated (multiplicity \(m\)): need extra factors of \(x^k\) (\(k=0\ldots m-1\)). Complex vs. real coefficients Real coefficients → complex roots appear in conjugate pairs, giving real sin/cos forms. Complex coefficients → keep exponential \(e^{(a\pm ib)x}\) directly. ⚠️ Common Misunderstandings “Homogeneous” means “no solution” – actually it means the right‑hand side is zero; solutions exist and form a vector space. Ignoring multiplicity – forgetting the \(x^k\) factors for repeated roots yields an incomplete basis. Applying undetermined coefficients to arbitrary \(f(x)\) – the method fails for functions like \(\ln x\) or \(\tan x\); use variation of parameters instead. Integrating factor sign – the factor is \(e^{\int f(x)dx}\), not \(e^{-\int f(x)dx}\) (the latter appears in the derivation step). Fundamental matrix assumption – \(U(x)=e^{\int A(x)dx}\) works only if \(A(x)\) commutes with its integral; otherwise more sophisticated methods are needed. 🧠 Mental Models / Intuition Characteristic polynomial = “frequency detector” – each root tells you a natural exponential (or oscillatory) mode that the system can sustain. Vector‑space view – homogeneous solutions are like “directions” you can move freely; the particular solution is a “shift” to satisfy the forcing. Integrating factor = “magic multiplier” that turns a non‑exact derivative into an exact one, akin to adding a weight to balance a scale. Cauchy–Euler scaling – the equation is scale‑invariant; substituting \(\ln x\) changes scaling into translation, which is easier to handle. 🚩 Exceptions & Edge Cases Zero leading coefficient – if \(an(x)\equiv0\) the declared order drops; ensure the highest non‑zero coefficient defines the order. Repeated root with complex multiplicity – still need \(x^k e^{\alpha x}\) factors, where \(\alpha\) is complex; the real form combines sin/cos with polynomial factors. Non‑commuting \(A(x)\) in systems – the simple exponential formula for \(U(x)\) fails; use Peano‑Baker series or compute a fundamental matrix by other means. Euler equation with \(x=0\) – substitution \(x=e^{t}\) requires \(x>0\); solutions may need separate treatment at \(x=0\). 📍 When to Use Which Constant coefficients & simple RHS → try undetermined coefficients first (fast). RHS is another solution of a homogeneous ODE or a holonomic function → use annihilator method. RHS arbitrary (e.g., \(\ln x\), \(\tan x\)) → apply variation of parameters. First‑order linear ODE → always use integrating factor (quadrature). Higher‑order ODE with variable coefficients → if it matches Euler form, use log‑substitution; otherwise resort to variation of parameters or numerical methods. Linear system with constant (or commuting) matrix → solve homogeneous part via matrix exponential, then use the integral formula for particular solution. 👀 Patterns to Recognize Discriminant sign → instantly tells you the nature of second‑order solutions (real distinct, repeated, or oscillatory). Right‑hand side contains a term already in the homogeneous basis → need to multiply the trial particular solution by \(x\) (or higher powers) to avoid duplication. Repeated root of multiplicity \(m\) → look for polynomial factor of degree \(m-1\) multiplying the exponential. Euler equations → coefficients follow powers of \(x\) descending by one each derivative; spot this pattern to switch to \(t=\ln x\). System matrix \(A(x)\) constant → solution is \(e^{Ax}\); if \(A\) is diagonalizable, exponentiate eigenvalues directly. 🗂️ Exam Traps Choosing wrong trial form – e.g., using \(Ae^{\beta x}\) when \(\beta\) is a root of the characteristic equation; the correct trial must be multiplied by \(x\). Forgetting the \(c1,c2\) constants – answer choices that omit arbitrary constants are incomplete. Sign error in integrating factor – a common distractor flips the exponent sign, leading to an incorrect particular solution. Misidentifying multiplicity – missing that a double root gives both \(e^{\alpha x}\) and \(x e^{\alpha x}\); answer choices may list only one. Assuming matrix exponential works for non‑commuting \(A(x)\) – some problems purposely give variable‑coefficient matrices that do not commute; the simple \(e^{\int A dx}\) answer will be wrong. Cauchy‑Euler domain – answer choices that present solutions valid only for \(x>0\) when the problem expects a full domain may be a trap; remember the log substitution requires \(x>0\).