Wronskian - Study guides, Class notes & Summaries

Chapter 2 Second-Order Differential Equations 2.1 The Linear Second-Order Equation 1. It is a routine exercise in differentiation to show that y1 (x) and y2 (x) are solutions of the homogeneous equation, while yp (x) is a solution of the nonhomogeneous equation. The Wronskian of y1 (x) and y2 (x) is W (x) = sin(6x) cos(6x) = −6 sin2 (x) − 6 sin2 (x) = −6, 6 cos(6x) −6 sin(6x) and this is nonzero for all x, s...

ay' + by = 0, integrating factor correct answers µ = exp(∫(b(t)/a(t))dt) ay' + by = h, variation of parameters correct answers Guess y_p=g(x)f(x), where f(x) is the homoegenous solution, and plug into l(y)=ay' + by = h. Result: g' = h(x)/[a(x)f(x)]. We have to INTEGRATE g' and MULTIPLY it by f. ay' + by = h, general solution correct answers homogeneous solution + particular solution Analytic definition correct answers If f is a smooth function, then f is analytic at x₀∈‖R...

Differential Equations Exam Questions with Correct Answers True or False? Every homogeneous linear differential equation always has a solution - Answer-True True or False? In the growth model dP/dt=50e^0.5t, the growth constant is 50 - Answer-False True or False? In the competition model dx/dt = ax-bxy, dy/dt=cy-dxy, where x(t) and y(t) are the popultions of the competing species moose and deer, respectively, the coefficient c represents the moose growth rate - Answer-False True or Fa...

2.1 The Linear Second-Order Equation 1. It is a routine exercise in differentiation to show that y1 (x) and y2 (x) are solutions of the homogeneous equation, while yp (x) is a solution of the nonhomogeneous equation. The Wronskian of y1 (x) and y2 (x) is W (x) = sin(6x) cos(6x) = −6 sin2 (x) − 6 sin2 (x) = −6, 6 cos(6x) −6 sin(6x) and this is nonzero for all x, so these solutions are linearly independent on the real line. ...

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MAT3706 Full study notes 2022 Chapter(1( ! ! x1 (t) = a11(t) x1 + a12 (t) x2 + :… + a1n(t) xn + Q1(t) - Qk(t) are called forcing terms - homogeneous when Qk (t) is zero for all k = 1; 2; … ; n - Non-homogenous if at least one Q k (t) is not zero - Solution is defined and differentiable Representation of homogenous system: Higher–order system with constant coefficients - Has polynomial differential operators Determinant of the system: Degenerate: If det = 0 Non–degenerate:...

AU MATH 376 STUDY GUIDE ORDINARY DIFFERENTIAL EQUATIONS NEW UPDATED SOLUTION FOR THE COMING 2022-2023 EXAM Athabasca University Contents Introduction 1 Some Preliminary Remarks 2 Part I: First-order Differential Equations 9 1 Introducing Ordinary Differential Equations 11 1.1 Solutions: Complete, General, Particular, Partial 12 1.2 Methods for Solving Ordinary Differential Equations 14 1.3 The Fundamental Theorem 15 1.4 Direction Fields 18 2 Directly Integrable Ordinary Differential ...

Math 246 Notes May 7, 2021 This note may contain some typos. Feel free to message me if you see any typos. Contents 1 Week 1 3 1.1 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Explicit First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 First-Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Summary . . . . . . . . . . . . . . . . . . . . . ....

DIFFERENTIAL EQUATIONS EXAM QUESTIONS WIT ALL CORRECT ANSWERS Two questions arise when approaching a DE - Answer-Does it have a solution and is it unique? In the logistic model for population growth dP/dt=P(8-2P), the carrying capacity of the population is - Answer-k=a/b so in this case, 4 What is the difference between an IVP and a BVP? - Answer-an initial value problem has all of the conditions specified at the same value of the independent variable in the equation. ex, y(0)= #, y'(...

Review of lesson 5.1 for MA 262 (Linear Algebra & Differential Equations) at Purdue University. Content based on class textbook Differential Equations & Linear Algebra by C. Henry Edwards and David E. Penney (ISBN: 9780536264923). 5.1 introduces the concept of 2nd-order linear differential equations, including definitions, general form, homogeneous equations vs. nonhomogeneous equations, the principle of superposition for homogeneous equations, existence and uniqueness of solutions, the Wronsk...