SlideServe
Télécharger Obtenir Pro
Menu
Accueil Pro
Préférences
ApparenceLight Aide et commentaires
© SlideServe · v2026

Higher-Order Differential Equations

cameron-rush avatar
cameron-rush
92 diapositives · 2.5K Views · Nov 18, 2014

CHAPTER 3. Higher-Order Differential Equations. (3.1~3.6). Chapter Contents. 3.1 Theory of Linear Equations 3.2 Reduction of Order 3.3 Homogeneous Linear Equations with Constants Coefficients 3.4 Undetermined Coefficients 3.5 Variation of Parameters 3.6 Cauchy-Euler Equations.

#higher-order differential equations #higher-order differential equations ppt #powerpoint #presentation
Higher-Order Differential Equations - slide 1
01 / 92Higher-Order Differential Equations
1 / 17

Télécharger la présentation

Higher-Order Differential Equations
Une image / un lien ci-dessous est fourni (tel quel) pour télécharger la présentation

Politique de téléchargement : le contenu du site vous est fourni TEL QUEL pour votre information et votre usage personnel ; il ne peut être vendu, concédé sous licence ni partagé sur d’autres sites sans l’accord de son auteur.Le contenu vous est fourni TEL QUEL, pour votre information et votre usage personnel uniquement. Téléchargez la présentation en cliquant sur ce lien. Lors du téléchargement, si vous ne parvenez pas à télécharger une présentation, l’éditeur a peut-être supprimé le fichier de son serveur.Pendant le téléchargement, si vous ne parvenez pas à obtenir une présentation, le fichier a peut-être été supprimé par l’éditeur.

F I N

Plus de contenu associé

Transcription de la présentation

  1. CHAPTER 3 Higher-Order Differential Equations (3.1~3.6)

  2. Chapter Contents • 3.1 Theory of Linear Equations • 3.2 Reduction of Order • 3.3 Homogeneous Linear Equations with Constants Coefficients • 3.4 Undetermined Coefficients • 3.5 Variation of Parameters • 3.6 Cauchy-Euler Equations

  3. 3.1 Preliminary Theory: Linear Equ. • Initial-value ProblemAn nth-order initial problem isSolve: Subject to:(1)with n initial conditions.

  4. Theorem 3.1.1 Existence of a Unique Solution Let an(x), an-1(x), …, a0(x),and g(x)be continuous on I, an(x)  0for all x on I. If x = x0 is any point in this interval, then a solution y(x)of (1) exists on the interval and is unique.

  5. Example 1 Unique Solution of an IVP • The IVP possesses the trivial solution y = 0. Since this DE with constant coefficients, from Theorem 3.1.1, hence y = 0 is the only one solution on any interval containing x = 1.

  6. Example 2 Unique Solution of an VP • Please verify y = 3e2x + e–2x– 3x,is a solution of This DE is linear and the coefficients and g(x)are all continuous, and a2(x)  0 on any I containing x = 0.This DE has an unique solution on I.

  7. Boundary-Value Problem • Solve:Subject to: is called a boundary-value problem (BVP).See Fig 3.1.1.

  8. Fig 3.1.1 Colored curves are solution of a BVP

  9. Example 3 A BVP Can Have Money, Ine, or Not Solutions • In example 4 of Sec 1.1, we see the solution of is x = c1 cos 4t + c2 sin4t (2) (a) Suppose x(0) = 0,then c1 = 0, x(t) = c2 sin 4t Furthermore, x(/2) = 0,we obtain 0 = 0,hence(3)has infinite many solutions. See Fig 3.1.2. (b) If(4)we have c1 = 0, c2= 0, x = 0is the only solution.

  10. Example 3 (2) (c) If (5)we have c1 = 0, and 1 = 0 (contradiction).Hence (5) has no solutions.

  11. Fig 3.1.2 The BVP in (3) of Ex 3 has many solutions

  12. Homogeneous Equations • The following DE(6)is said to be homogeneous;(7)with g(x)not identically zero, is nonhomogeneous.

  13. Differential Operators • Let dy/dx = Dy. This symbol D is called a differential operator. We define an nth-order differential operator as (8)In addition, we have(9)so the differential operator L is a linear operator. • Differential EquationsWe can simply write the DEs as L(y) = 0andL(y) = g(x)

  14. Let y1, y2, …, ykbe a solutions of the homogeneous Nth-order differential equation (6) on an interval I. Then the linear combinationy = c1y1(x) + c2y2(x)+ …+ ckyk(x)where the ci, i = 1, 2, …, k are arbitrary constants, is also a solution on the interval. Theorem 3.1.2 Superposition Principles – Homogeneous Equations

  15. Corollaries to Theorem 3.1.2 (a) y = cy1 is also a solution if y1is a solution. (b) A homogeneous linear DE always possesses the trivial solution y = 0.

  16. A set of f1(x), f2(x), …, fn(x)is linearly dependent on an interval I, if there exists constants c1, c2, …, cn, not all zero, such thatc1f1(x) + c2f2(x) + … + cn fn(x) = 0If not linearly dependent, it is linearly independent. Definition 3.1.1 Linear Dependence / Independence Example 4 Superposition – Homogeneous DE • The function y1= x2, y2= x2 ln x are both solutions of Then y = x2+ x2 ln x is also a solution on (0, ).

  17. In other words, if the set is linearly independent,when c1f1(x) + c2f2(x) + … + cn fn(x) = 0then c1 = c2 = … = cn= 0 • Referring to Fig 3.1.3, neither function is a constant multiple of the other, then these two functions are linearly independent. • If two functions are linearly dependent, then one is simply a constant multiple of the other. • Two functions are linearly independent when neither is a constant multiple of the other.

  18. Fig 3.1.3 The set consisting of f1 and f2 is linear independent on (-, )

  19. Example 5 Linear Dependent Functions • The functions f1 = cos2 x, f2 = sin2x, f3 = sec2x, f4 =tan2x are linearly dependent on the interval (-/2, /2) sincec1 cos2 x +c2 sin2x +c3 sec2x +c4 tan2x = 0when c1 = c2 = 1, c3 = -1, c4 = 1.

  20. Example 6 Linearly Dependent Functions • The functions are linearly dependent on the interval (0, ), sincef2 = 1 f1 + 5 f3 + 0 f4

  21. Suppose each of the functions f1(x), f2(x), …, fn(x) possesses at least n – 1 derivatives. The determinantis called the Wronskianof the functions. Definition 3.1.2 Wronskian

  22. Let y1(x), y2(x), …, yn(x) be n solutions of the nth-order homogeneous DE (6) on an interval I. This set of solutions is linearly independent on I if and on if W(y1, y2, …, yn)  0 for every x in the interval. Theorem 3.1.3 Criterion for Linear Independence Solutions

  23. Definition 3.1.3 Fundamental Set of Solutions Any set y1(x), y2(x),… , yn(x) of n linearly independent solutions is said to be a fundamentalset of solutions. There exists a fundamental set of solutions for (6) on an interval I. Theorem 3.1.4 Existence of a Fundamental Set

  24. Let y1(x), y2(x), …, yn(x) be a fundamental set of solutions of homogeneous DE (6) on an interval I. Then the general solution isy = c1y1(x) + c2y2(x) + … + cnyn(x) where ci, i = 1, 2, …, n are arbitrary constants. Theorem 3.1.5 General Solution – Homogeneous Equations

  25. Example 7 General Solution of a Homogeneous DE • The functions y1 = e3x, y2 = e-3xare solutions of y” –9y = 0on (-, )Now for every x. So y = c1e3x + c2e-3xis the general solution.

  26. Example 8 A Solution Obtained from a General Solution • The functions y = 4sinh 3x - 5e3x is a solution of example 7 (Verify it). Observer y = 4 sinh 3x – 5e-3x

  27. Example 9 General Solution of a Homogeneous DE • The functions y1= ex,y2 = e2x , y3 = e3xare solutions of y’’’ –6y” + 11y’ –6y = 0on (-, ). Sincefor everyrealvalueof x. So y = c1ex + c2e2x + c3e3xis the general solution on (-, ).

  28. Any ypfree of parameters satisfying (7) is called a particular solution. If y1(x), y2(x), …, yk(x) be a fundamental set of solutions of (6), then the general solution of (7) isy= c1y1 + c2y2 +… +ckyk + yp(10)where ci, i = 1, 2, …, n are arbitrary constants. Theorem 3.1.6 General Solution – Nonhomogeneous Equations Nonhomogeneous Equations • Complementary Functiony = c1y1 + c2y2 +… +ckyk +yp = yc + yp = complementary + particular

  29. Example 10 General Solution of a Nonhomogeneous DE • The function yp= -(11/12) –½ xis a particular solution of (11)From previous discussions, the general solution of (11) is

  30. Given(12)where i = 1, 2, …, k. If ypi denotes a particular solution corresponding to the DE (12) with gi(x), then(13)is a particular solution of (14) Theorem 3.1.7 Superposition Principles – Nonhomogeneous Equations Any Superposition Principles

  31. Example 11 Superposition – Nonmogeneous DE • We findis a particular solution ofis a particular solution ofis a particular solution of From Theorem 3.1.7, is a solution of

  32. Note • If ypi is a particular solution of (12), then is also a particular solution of (12) when the right-hand member is

  33. 3.2 Reduction of Order • IntroductionWe know the general solution of (1)is y = c1y1 + c2y1. Suppose y1(x)denotes a known solution of (1). We assume the other solution y2has the form y2= uy1.Our goal is to find a u(x)and this method is called reduction of order.

  34. Example 1 Finding s second Solution Given y1 = ex is a solution of y” – y = 0,find a second solution y2 by the method of reduction of order. Solution:If y = uex, then And Since ex  0, we let w = u’,then

  35. Example 1 (2) Thus(2)Choosing c1= 0, c2= -2, we have y2= e-x. Because W(ex, e-x)  0for every x, they are independent on (-, ).

  36. General Case • Rewrite (1) as the standard form(3)Let y1(x)denotes a known solution of (3) and y1(x)  0 for every x in the interval. • If we define y = uy1, thenwe have

  37. This implies that or(4)where we let w = u’. Solving (4), we have or

  38. thenLet c1 = 1, c2 = 0,we find (5)

  39. Example 2 A second Solution by Formula (5) The function y1= x2is a solution of Find the general solution on (0, ). Solution:The standard form is From (5)The general solution is

  40. 3.3 Homogeneous Linear Equation with Constant Coefficients • Introduction(1)where ai, i = 0, 1, …, n are constants, an 0. • Auxiliary EquationFor n = 2,(2)Try y = emx, then (3)is called an auxiliary equation.

  41. From (3) the two roots are(1) b2 – 4ac > 0: two distinct real numbers.(2) b2 – 4ac = 0: two equal real numbers.(3) b2 – 4ac < 0: two conjugate complex numbers.

  42. Case 1: Distinct real rootsThe general solution is(4) • Case 2: Repeated real rootsand from (5) of Sec 3.2, (5)The general solution is (6)

  43. Case 3: Conjugate complex rootsWe write , a general solution is From Euler’s formula: and(7) and

  44. Sinceis a solution then setC1= C1 = 1and C1 = 1, C2 = -1 , we have two solutions: So, ex cos x and exsin x are a fundamental set of solutions, that is, the general solution is(8)

  45. Example 1 Second-Order DEs • Solve the following DEs: (a) (b) (c)

  46. Example 2 An Initial-Value Problem Solve Solution:See Fig 3.3.1.

  47. Fig 3.3.1 Graph of a solution of IVP in Ex 2

  48. Two Equations worth Knowing • For the first equation:(9)For the second equation:(10)Let Then(11)

  49. Higher-Order Equations • Given (12)we have(13)as an auxiliary equation. • If the roots of (13) are real and distinct, then the general solution of (12) is

  50. Example 3 Third-Order DE Solve Solution: