Ordinary differential equation Study Guide

📖 Core Concepts Ordinary Differential Equation (ODE): Relates an unknown function \(y(x)\) to its derivatives with respect to a single variable \(x\). Linear ODE: The unknown function and all its derivatives appear linearly; coefficients may be any differentiable functions of \(x\). Explicit vs. Implicit ODE: Explicit – the highest‑order derivative \(y^{(n)}\) stands alone: \(F(x,y,y',\dots ,y^{(n)})=0\). Implicit – the highest‑order derivative is mixed with other terms: \(G(x,y,y',\dots ,y^{(n)})=0\). Solution Types: General solution – contains \(n\) arbitrary constants for an \(n\)th‑order ODE. Particular solution – obtained by fixing constants to satisfy initial/boundary conditions. Singular solution – cannot be produced by any choice of constants in the general solution. Homogeneous solution – solves the associated equation with the non‑homogeneous term set to zero. Existence & Uniqueness (Picard–Lindelöf): If \(f(x,y)\) and \(\partial f/\partial y\) are continuous (and Lipschitz in \(y\)) on a rectangle containing \((x0,y0)\), a unique local solution to \(y'=f(x,y),\; y(x0)=y0\) exists; it can be extended globally if the hypotheses hold on the whole domain. --- 📌 Must Remember Linear ODE definition: \(\displaystyle an(x)y^{(n)}+\dots +a1(x)y'+a0(x)y = b(x)\). Integrating factor for first‑order linear ODE: \(\mu(x)=e^{\int p(x)\,dx}\). Characteristic polynomial (constant coefficients): \[ a0\lambda^{n}+a1\lambda^{n-1}+\dots +an=0 \] – each distinct real root \(\lambdak\) → term \(e^{\lambdak x}\); complex conjugate roots \(\alpha\pm i\beta\) → \(e^{\alpha x}\bigl(C1\cos\beta x + C2\sin\beta x\bigr)\). Repeated root \(\lambda\) of multiplicity \(m\) → multiply by \(x^{0},x^{1},\dots ,x^{m-1}\). Variation of parameters: For \(y''+p(x)y'+q(x)y=r(x)\), write \(yp = u1(x)y1+u2(x)y2\) and solve \[ \begin{cases} u1' y1 + u2' y2 = 0\\ u1' y1' + u2' y2' = r(x) \end{cases} \] Method of undetermined coefficients: Guess a form for \(yp\) mirroring \(r(x)\) (polynomial, exponential, sinusoid) and solve for the unknown coefficients; adjust by multiplying by \(x\) if the guess duplicates a homogeneous term. Laplace transform: \(\mathcal{L}\{y'(t)\}=sY(s)-y(0)\); converts constant‑coefficient linear ODEs to algebraic equations in \(s\). Picard–Lindelöf conditions: Continuity of \(f\) and Lipschitz continuity in \(y\) → unique local solution. --- 🔄 Key Processes Separate Variables (first‑order separable): Write \(\displaystyle \frac{dy}{dx}=g(x)h(y)\). Rearrange: \(\displaystyle \frac{1}{h(y)}\,dy = g(x)\,dx\). Integrate both sides, add constant \(C\). Solve First‑Order Linear ODE: Standard form: \(y' + p(x)y = q(x)\). Compute integrating factor \(\mu(x)=e^{\int p(x)dx}\). Multiply equation by \(\mu\): \((\mu y)' = \mu q(x)\). Integrate: \(y = \mu^{-1}\bigl(\int \mu q(x)dx + C\bigr)\). Reduction to First‑Order System (order \(n\) → \(n\) equations): Set \(y1 = y,\; y2 = y',\dots , yn = y^{(n-1)}\). Write \(\displaystyle yi' = y{i+1}\) for \(i=1,\dots,n-1\); the last equation comes from the original ODE. Variation of Parameters (second‑order): Find two independent homogeneous solutions \(y1, y2\). Solve for \(u1', u2'\) using the system above. Integrate to get \(u1, u2\); assemble \(yp = u1 y1 + u2 y2\). Method of Undetermined Coefficients: Identify the form of \(r(x)\) (e.g., \(e^{ax}\), \(\sin bx\), polynomial). Propose \(yp\) with undetermined constants of the same form. If any term duplicates a homogeneous term, multiply the guess by \(x\). Substitute into the ODE, solve for the constants. Laplace Transform Solution: Take \(\mathcal{L}\) of each term, use initial conditions to handle derivatives. Solve the resulting algebraic equation for \(Y(s)\). Find \(y(t)\) by inverse Laplace transform (tables or partial fractions). Euler’s Method (numerical): Starting at \((x0,y0)\) with step \(h\): \[ y{k+1}=yk + h\,f(xk,yk) \] Iterate to approximate the solution on the interval. --- 🔍 Key Comparisons Explicit vs. Implicit ODE Explicit: highest derivative isolated → easier to solve analytically. Implicit: derivative mixed → may require algebraic manipulation or numerical methods. Homogeneous vs. Particular Solution Homogeneous: solves \(L[y]=0\); forms the “basis” of the solution space. Particular: solves \(L[y]=r(x)\); added to homogeneous to satisfy non‑homogeneous ODE. Separable vs. Linear First‑Order Separable: can be written as \(g(x)h(y)\); solved by direct integration. Linear: has form \(y'+p(x)y=q(x)\); solved via integrating factor. Undetermined Coefficients vs. Variation of Parameters Undetermined Coefficients: fast but limited to RHS that are exponentials, polynomials, sines/cosines. Variation of Parameters: universal (works for any \(r(x)\)) but more algebraic work. Euler vs. Runge–Kutta Euler: first‑order, simple, larger truncation error. Runge–Kutta (e.g., RK4): higher‑order, more accurate for same step size. --- ⚠️ Common Misunderstandings “Any linear ODE has a closed‑form solution.” Only constant‑coefficient linear ODEs are guaranteed to be solvable by elementary functions; variable coefficients often require series or numerical methods. Confusing “particular” with “singular” solutions. A particular solution comes from the general solution; a singular solution cannot be obtained this way. Dropping the constant of integration in separable equations. Always add \(+C\); forgetting it loses the family of solutions. Using the integrating factor incorrectly (sign error). The factor is \(e^{\int p(x)dx}\); a missing minus sign flips the whole solution. Assuming continuity of \(f\) alone guarantees global uniqueness. Lipschitz continuity (or a stronger condition) is required for uniqueness; continuity alone gives only existence. Resonance in undetermined coefficients: When the guessed form already appears in the homogeneous solution, you must multiply by \(x\) (or higher powers) to obtain a valid particular ansatz. --- 🧠 Mental Models / Intuition Superposition: For linear ODEs, think of the solution as building blocks (homogeneous modes) plus a forced response (particular). Characteristic roots = “natural frequencies.” Real roots → exponential decay/growth; complex roots → damped oscillations. Variation of parameters = “let the constants breathe.” The constants become functions that adjust to satisfy the forcing term. Phase portrait = “vector field snapshot.” In a 2‑D system, trajectories flow along arrows; fixed points = equilibrium. Laplace transform = “algebraic cheat sheet.” Derivatives become multiplication by \(s\), turning differential problems into algebraic ones. --- 🚩 Exceptions & Edge Cases Repeated characteristic roots: Multiply the exponential term by \(x, x^2,\dots\) according to multiplicity. Complex conjugate roots: Produce sinusoidal terms; remember Euler’s formula \(e^{i\beta x} = \cos\beta x + i\sin\beta x\). Regular singular points: Ordinary power‑series may fail; use Frobenius method (series with possibly non‑integer powers). Non‑Lipschitz RHS: Example \(y' = \sqrt{|y|}\) at \(y=0\) – multiple solutions may exist. Resonance: For RHS matching a homogeneous term (e.g., \(r(x)=e^{\lambda x}\) when \(\lambda\) is a root), the particular ansatz must be multiplied by \(x\). --- 📍 When to Use Which | Situation | Recommended Method | |-----------|---------------------| | ODE of form \(\frac{dy}{dx}=g(x)h(y)\) | Separate variables | | First‑order linear \(y'+p(x)y=q(x)\) | Integrating factor | | Constant‑coefficient linear ODE (any order) | Characteristic polynomial (solve for roots) | | RHS is exponential/polynomial/trigonometric & no resonance | Undetermined coefficients | | RHS not covered by simple ansatz (e.g., product of functions) | Variation of parameters | | Initial‑value problem with constant coefficients, need quick solution | Laplace transform (handles ICs automatically) | | Variable coefficients, no closed form | Series (Frobenius) or numerical integration | | System of first‑order equations, want qualitative picture | Phase portrait or numerical solver | | Need high accuracy over large interval | Runge–Kutta (RK4) rather than Euler | --- 👀 Patterns to Recognize Repeated root pattern: Characteristic polynomial yields \((\lambda-\lambda0)^m\) → expect terms \(e^{\lambda0 x}, x e^{\lambda0 x}, \dots , x^{m-1}e^{\lambda0 x}\). Complex root pattern: Coefficients of sine and cosine appear together; check if the ODE has constant coefficients and a negative discriminant. RHS = polynomial × \(e^{ax}\): Guess a polynomial of same degree times \(e^{ax}\); if \(a\) is a root of the characteristic equation, multiply by \(x\). RHS = sinusoid: Guess \(A\sin(bx)+B\cos(bx)\); if \(\pm ib\) are characteristic roots, multiply by \(x\). Separable form hidden: Look for \(\frac{dy}{dx}=g(x)h(y)\) after algebraic rearrangement (e.g., \(\frac{y'}{y}=k(x)\)). Linear system with constant matrix: Eigenvalues of \(\mathbf{A}\) give exponential modes; complex eigenvalues give spirals in phase portrait. --- 🗂️ Exam Traps Missing the constant of integration in separable or series solutions → answer lacks the full family. Wrong sign in integrating factor (using \(e^{-\int p(x)dx}\)) → leads to incorrect particular solution. Assuming a particular solution form works without checking resonance – you’ll obtain a solution that actually satisfies the homogeneous equation, leaving the ODE unsatisfied. Confusing Laplace transform of a derivative: Remember \(\mathcal{L}\{y''\}=s^2Y(s)-sy(0)-y'(0)\). Forgetting the initial terms yields a wrong algebraic equation. Choosing Euler’s method with too large a step – the error may be dramatic, but the answer still looks plausible; exam may ask for error estimate. Treating a non‑Lipschitz RHS as guaranteeing uniqueness – can produce multiple valid solutions; exam may test with an example like \(y' = |y|^{1/2}\). Overlooking the need to convert a higher‑order ODE to a first‑order system before applying numerical solvers that accept only first‑order form. ---