Section 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
|
|
- Robyn Carol McKenzie
- 5 years ago
- Views:
Transcription
1 Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1
2 General Results THEOREM 1: If z = z 1 (x) and z = z 2 (x) are solutions of equation (N), then y(x) = z 1 (x) z 2 (x) is a solution of equation (H). Proof: 2
3 THEOREM 2: Let y = y 1 (x) and y = y 2 (x) be linearly independent solutions of the reduced equation (H) and let z = z(x) be a particular solution of (N). Then y(x) = C 1 y 1 (x) + C 2 y 2 (x) + z(x) is the general solution of (N). 3
4 The general solution of (N) consists of the general solution of the reduced equation (H) plus a particular solution of (N): y = C 1 y 1 (x) + C 2 y 2 (x) }{{} + z(x). }{{} gen soln (H) + part soln (N) 4
5 To find the general solution of (N) you need to find: (i) a fundamental set of solutions y 1, y 2 of the reduced equation (H), and (ii) a particular solution z of (N). 5
6 THEOREM 3: (Superposition Principle) Given the nonhomogeneous equation y + p(x)y + q(x)y = f(x) + g(x). If z = z f (x) is a particular solution of y + p(x)y + q(x)y = f(x) and z = z g (x) is a particular solution of y + p(x)y + q(x)y = g(x), 6
7 then z(x) = z f (x) + z g (x) is a particular solution of y + p(x)y + q(x)y = f(x) + g(x). Proof:
8 Variation of Parameters Let y = y 1 (x) and y = y 2 (x) be independent solutions of the reduced equation (H) and let W(x) = y 1 y 2 y 1 y 2 = y 1y 2 y 2y 1 0 be their Wronskian. Then y = C 1 y 1 (x) + C 2 y 2 (x) is the general solution of (H). 7
9 Set z(x) = u(x)y 1 (x) + v(x)y 2 (x) where u and v are to be determined so that z is a solution of (N). z = uy 1 + vy 2 z = uy 1 + y 1u + vy 2 + y 2v Set y 1 u + y 2 v = 0 so we have z = uy 1 + vy 2 z = uy 1 + vy 2 and z = uy 1 + y 1 u + vy 2 + y 2 v
10 Substitute z, z, z into the differential equation ( uy 1 + y 1 u + vy 2 + y 2 v ) +p ( uy 1 + ) vy 2 + q (uy 1 + vy 2 ) = f Rearrange the terms: u ( y 1 + py 1 + qy ) ( 1 + v y 2 + py 2 + qy ) 2 + y 1 u + y 2 v = f 8
11 We now have two equations in the two unknowns u and v : y 1 u + y 2 v = 0 y 1u + y 2v = f Solve for u : ( y1 y 2 y 2y 1 ) u = y 2 f or Wu = y 2 f so u = y 2f W and u = y 2 f W dx; 9
12 y 1 u + y 2 v = 0 y 1u + y 2v = f Solve for v : ( y1 y 2 y 2y 1 ) v = y 1 f or Wv = y 1 f so v = y 1f W and v = y 1 f W dx; 10
13 Therefore, z(x) = y 1 (x) y 2 (x)f(x) W(x) dx+y 2 (x) y 1 (x)f(x) W(x) dx is a particular solution of the nonhomogeneous equation (N). 11
14 Examples: 1. {y 1 (x) = x 2, y 2 (x) = x 4 } is a fundamental set of solutions of y 5 x y + 8 x 2 y = 0. Find a particular solution z of the equation. y 5 x y + 8 x 2 y = 4x3 12
15 y 1 = x 2, y 2 = x 4 W[y 1, y 2 ] = 13
16 2. Find the general solution of y 4y + 4y = e2x x 14
17
18 3. Find the general solution of y + y 6y = 3e 2x 15
19
20 4. Find the general solution of y + 4y = 2tan2x 16
21
22
23
24
25
26
27 Section 3.5. Undetermined Coefficients aka Guessing NOTE: This method can be used only when: 1. the de has constant coefficients 2. f is an exponential function y + ay + by = f(x) a, b constants, f an exponential function. 17
28 Case 1: If y + ay + by = ce rx Set z(x) = Ae rx Example 1: Find a particular solution of y 5y + 6y = 7e 4x. Set z = Ae 4x where A is to be determined. 18
29 Answer: z = 1 6 e 4x. The general solution of the differential equation is: y = C 1 e 2x + C 2 e 3x e 4x. Note: If L[y] = y + ay + by, then L[Ae rx ] = A ( r 2 + ar + b ) e rx = Ke rx That is, L[Ae rx ] is a constant multiple of e rx 19
30 Example 2: Find a particular solution of y + 2y 3y = 9e 2x. 20
31 Case 2: y + ay + by = c cos βx, or y + ay + by = c sinβx, or y +ay +by = c cos βx+d sinβx, Example: y 2y + y = 5cos 2x 21
32 Note: If L[y] = y + ay + by, then L[Acos βx] = K cos βx + M sin βx That is, L[Acos βx] involves BOTH cosine and sine. Similarly for L[B sin βx] and L[Acos βx + B sinβx] Therefore, if f(x) = ccos βx or f(x) = dsin βx or f(x) = c cos βx + d sin βx 22
33 set z(x) = A cos βx + B sin βx where A, B are to be determined 23
34 Example 3: Find a particular solution of y 2y + y = 5cos 2x. Set z = Acos 2x + B sin 2x 24
35 Answer: z = 3 5 cos 2x 4 5 sin 2x. The general solution of the differential equation is: y = C 1 e x + C 2 xe x 3 5 cos 2x 4 5 sin 2x. 25
36 Example 4: Find a particular solution of y 2y + 5y = 2cos 3x 4sin 3x Set z = Acos 3x + B sin 3x + Ce 2x where A, B, C are to be determined. 26
37 y 2y + 5y = 2cos 3x 4sin 3x 27
38 Case 3: If f(x) = ce αx cos βx, de αx sin βx or ce αx cos βx + de αx sin βx set z(x) = Ae αx cos βx + Be αx sin βx where A, B are to be determined. 28
39 Example 5: Find a particular solution of y + 9y = 4e x sin 2x. Set z = Ae x cos 2x + Be x sin 2x Answer: z = 4 13 ex cos 2x ex sin 2x. 29
40 Example 6: Find a particular solution z of y 2y + y = 3e 3x 5sin2x. 30
41
42 Example 7: Find a particular solution z of y + y 6y = 3e 2x. 31
43 A BIG Difficulty: The trial solution z is a solution of the reduced equation. In this case, y 1 = e 3x and y 2 = e 2x are solutions of the reduced equation y + y 6y = 0 From Example 3, Section 3.4: z = 3 5 x e2x 32
44 Example 8: Find a particular solution z of y + y 6y = 3e 2x. Set z = Ae 2x? NO this satisfies the reduced equation, Set z = Axe 2x 33
45 Example 9: Find a particular solution of y 2y 15y = 6e 3x 34
46
47 Example 10: Find a particular solution z of y + 4y = 2cos 2x. 35
48
49 Example 11: Find a particular solution z of y + 6y + 9y = 4e 3x. 36
50
51 Example 12: Find a particular solution z of y 2y 8y = 3e 2x
52
53 Example 13: Find a particular solution z of y 3y = 4e 3x
54
55 Answers: z 9 = 3 4 x e 3x z 10 = 1 2 xsin 2x z 11 = 2x 2 e 3x z 12 = 1 2 xe 2x 3 4 z 13 = 4 3 x e3x 2 3 x 39
56 The Method of Undetermined Coefficients A. Applies only to equations of the form y + ay + by = f(x) where a, b are constants and f is an exponential function. c.f Variation of Parameters 40
57 B. Basic Case: If: f(x) = ae rx set z = Ae rx. f(x) = ccos βx, dsin βx, or ccos βx + dsin βx, set z = Acos βx + B sin βx. f(x) = ce αx cos βx, de αx sin βx or ce αx cos βx + de αx sin βx, set z = Ae αx cos βx + Be αx sin βx. 41
58 BUT: If z satisfies the reduced equation, use xz; if xz also satisfies the reduced equation, then x 2 z will give a particular solution. 42
59
60
61
62
63
64 C. General Case: If f(x) = p(x)e rx where p is a polynomial of degree n, then set z = P(x)e rx where P is a polynomial of degree n with undetermined coefficients. 43
65 Example 14: Find a particular solution of y 2y 8y = (4x + 5)e 2x. Set z = (Ax + B)e 2x. 44
66
67 Example 15: Find a particular solution of y 3y + 2y = (2x 2 1)e x. Set z = (Ax 2 + Bx + C)e x. Answer: z = ( 1 3 x x ) e x. 45
68
69 If f(x) = p(x)cos βx + q(x)sin βx where p, q are polynomials, then set z = P(x)cos βx + Q(x)sin βx where P, Q are polynomials of degree n with undetermined coefficients, n = max degree of p and q. 46
70 Example 16: y 2y 3y = 3 cos x + (x 2)sin x. Set z = (Ax+B)cos x+(cx+d)sin x. 47
71
72 Answer: z = ( 1 10 x 47 ) ( 1 cos x 50 5 x 2 ) sin x
73 If f(x) = p(x)e αx cos βx + q(x)e αx sin βx where p, q are polynomials of degree n, then set z = P(x)e αx cos βx + Q(x)e αx sin βx where P, Q are polynomials of degree n with undetermined coefficients. 49
74 Example 17: y +4y = 2x e x cos x. Set z = (Ax+B)e x cos x+(cx+d)e x sin x Answer: z = 1 25 (10x 7)ex cos x (5x 1)ex sin x. 50
75 Example 18: Find a particular solution of y 2y 8y = (4x + 5)e 2x. Set z = (Ax + B)e 2x???? 51
76 BUT: Warning!!! If any part of z satisfies the reduced equation, try xz; if any part of xz also satisfies the reduced equation, then x 2 z will give a particular solution. 52
77
78 Examples: 1. Give the form of a particular solution of y 4y 5y = 2cos 3x 5e 5x + 4. Answer: z = Acos 3x + B sin 3x + Cxe 5x + D 53
79 Answer: z = Ax + B + Cx 2 e 4x Give the form of a particular solution of y + 8y + 16y = 2x 1 + 7e 4x.
80 C cos 2x + Dsin 2x + Ee 2x Give the form of a particular solution of y + y = 4sin x cos 2x + 2e 2x. Answer: z = Axcos x + Bxsin x +
81 4. Give the form of the general solution of y + 9y = 4cos 2x + 3sin 2x Answer: y = C 1 cos 3x + C 2 sin 3x + Acos 2x + B sin 2x 56
82 5. Give the form of a particular solution of y + 9y = 4cos 3x + 3sin 2x Answer: z = Axcos 3x + Bxsin 3x + C cos 2x + Dsin 2x 57
83 6. Give the form of the general solution of y + 4y + 4y = 4xe 2x + 3 Answer: y = C 1 e 2x + C 2 xe 2x + (Ax 3 + Bx 2 )e 2x + C 58
84 7. Give the form of the general solution of y + 4y + 4y = 4e 2x sin 2x + 3x Answer: y = C 1 e 2x + C 2 xe 2x + Ae 2x cos 2x + Be 2x sin 2x + Cx + D 59
85 8. Give the form of the general solution of y + 4y = 4sin 2x + 3 Answer: y = C 1 e 4x +C 2 +Acos 2x+B sin 2x+Cx 60
86 9. Give the form of a particular solution of y + 2y + 10y = 2e 3x sin x + 4e 3x Answer: z = Ae 3x cos x + Be 3x sin x + Ce 3x 61
87 10. Give the form of the general solution of y + 2y + 10y = 2e x sin 3x + 2e x Answer: y = C 1 e x cos 3x+C 2 e x sin 3x+ Axe x cos 3x + Bxe x sin 3x + Ce x 62
88 11. Give the form of a particular solution of y 2y 8y = 2cos 3x (3x+1)e 2x 4 Answer: z = Acos 3x+B sin 3x+(Cx 2 +Dx)e 2x +E 63
89 12. Give the form of a particular solution of y 2y 8y = 2cos 3x 3xe 2x 3x Answer: z = Acos 3x+B sin 3x+(Cx 2 +Dx)e 2x +Ex + F 64
90 13. Find the general solution of y 4y + 4y = 4e 2x + e2x x Answer: y = C 1 e 2x + C 2 xe 2x 2x 2 e 2x + xe 2x ln x 65
91 Summary: 1. Variation of parameters: Can be applied to any linear nonhomogeneous equations, but requires a fundamental set of solutions of the reduced equation. 66
92 2. Undetermined coefficients: Is limited to linear nonhomogeneous equations with constant coefficients, and f must be an exponential function, f(x) = ae rx, f(x) = ccos βx + dsin βx, f(x) = ce αx cos βx + de αx sin βx, or p(x)f(x) p a polynomial. 67
93 In cases where both methods are applicable, the method of undetermined coefficients is usually more efficient and, hence, the preferable method. 68
94
95
96 Section 3.6. Vibrating Mechanical Systems 69
97 I. Free Vibrations Hooke s Law: The restoring force of a spring is proportional to the displacement: F = ky, k > 0. Newton s Second Law: Force equals mass times acceleration: F = ma = m d2 y dt 2. Mathematical model: m d2 y dt 2 = ky 70
98 which can be written where ω = k/m. d 2 y dt 2 + ω2 y = 0 ω is called the natural frequency of the system.
99 The general solution of this equation is: y = C 1 sin ωt + C 2 cos ωt which can be written y = Asin(ωt + φ). A is called the amplitude, φ is called the phase shift. 71
100 y = C 1 sin ωt + C 2 cos ωt 72
101 y = Asin(ωt + φ) A y t A 73
102 II. Forced Free Vibrations Apply an external force G to the freely vibrating system Force Equation: F = ky + G. Mathematical Model: my = ky+g or y + k m y = G m, a nonhomogeneous equation. 74
103
104 A periodic external force: G = a cos γt, a, γ > 0 const. Force Equation: F = ky + a cos γt Mathematical Model: y + k m y = a m cos γt y + ω 2 y = α cos γt where ω = k/m, α = a m. 75
105 ω is called the natural frequency of the system, γ is called the applied frequency. 76
106 Case 1: γ ω. y + ω 2 y = α cos γt General solution, reduced equation: y = C 1 cos ωt+c 2 sin ωt = Asin(ωt+φ 0 ). Form of particular solution (undetermined coefficients): z = Acos γt + B sin γt. A particular solution: z = α ω 2 γ 2 cos γt. 77
107 General solution: y = Asin(ωt + φ 0 ) + α ω 2 cos γt. γ2
108 ω/γ rational: periodic motion 1 ω/γ irrational: not periodic y 1 t 78
109 Case 2: γ = ω. y + ω 2 y = αcos ωt General solution, reduced equation: y = C 1 cos ωt + C 2 sin ωt = Asin(ωt + φ 0 ). Form of particular solution (undetermined coefficients): z = A tcos γt + B tsin γt. 79
110 A particular solution: α 2ω t sin ωt. General solution: y = Asin(ωt + φ 0 ) + α 2ω t sin ωt. Unbounded oscillation This is known as resonance 80
111 y = Asin(ωt + φ 0 ) + α 2ω t sin ωt. y t 81
112 III. Damped Free Vibrations: A resistance force R (e.g., friction) proportional to the velocity v = y and acting in a direction opposite to the motion: R = cy with c > 0. Force Equation: F = ky(t) cy (t). Newton s Second Law: F = ma = my 82
113 Mathematical Model: my (t) = ky(t) cy (t) or y + c m y + k m y = 0 (c, k, m constant) or y + αy + βy = 0 α = c m, β = k m α, β positive constants.
114 Characteristic equation: r 2 + α r + β = 0. Roots r = α ± α 2 4β. 2 There are three cases to consider: α 2 4β < 0, α 2 4β > 0, α 2 4β = 0. 83
115 Case 1: α 2 4β < 0. Complex roots: (Underdamped) r 1 = α 2 + iω, r 2 = α 2 iω where ω = 4β α 2. 2 General solution: y = e ( α/2)t (C 1 cos ωt + C 2 sin ωt) 84
116 or y(t) = A e ( α/2)t sin(ωt + φ 0 ) where A and φ 0 are constants, NOTE: The motion is oscillatory AND y(t) 0 as t. 85
117 Underdamped Case: y t 86
118 Case 2: α 2 4β > 0. Two distinct real roots: (Overdamped) r 1 = α + α 2 4β, r 2 = α α 2 4β. 2 2 General solution: y(t) = y = C 1 e r 1t + C 2 e r 2t. The motion is nonoscillatory. 87
119 NOTE: Since α 2 4β < α 2 = α, r 1 and r 2 are both negative and y(t) 0 as t. 88
120 Case 3: α 2 4β = 0. One real root: (Critically Damped) r 1 = r 2 = α 2, General solution: y(t) = y = C 1 e (α/2) t + C 2 t e (α/2) t. The motion is nonoscillatory and y(t) 0 as t. 89
121 Overdamped and Critically Damped Cases: y y t t y t 90
122 Summary of Case III: All solutions of y + ay + by = 0 have limit 0 as t. That is, in the presence of a resistant force (friction), all solutions ultimately return to the equilibrium position. 91
123 IV. Forced Damped Vibrations Apply an external force G to a damped, freely vibrating system Force Equation: F = ky cy + G. Mathematical Model: my = ky cy + G or y + c m y + k m y = G m, 92
124 which we write as y + αy + βy = g, where α = c/m, β = k/m, g = G/m A periodic external force: g = a cos γt, a, γ > 0 const. Mathematical Model: y + αy + βy = acos γt 93
125 General solution: y(t) = C 1 y 1 (t) + C 2 y 2 (t) + Z(t) = Y c (t) + Z(t), Note: From Case III lim t Y c(t) = 0. as t so lim t y(t) = Z(t). 94
126 Particular solution of (N) y + α y + β y = a cos γt will have the form: Z(t) = Acos γt + B sin γt. General solution of (N): y(t) = Y c (t) + Z(t) Note: lim y(t) = Z(t) t 95
127 Y c (t), the general solution of the reduced equation, is called the transient solution. Z(t) the particular solution of (N), is called the steady state solution. 96
128 Example 1: y y y = cos t General solution y = C 1 e t/4 +C 2 e t/ cos t sin t Transient solution: y(t) = 2e t/4 +e t/2 (C 1 = 2, C 2 = 1) Steady-state solution: Z(t) = cos t sin t 97
129 Transient solution: Steady-state solution: - 98
130 y = 2e t/4 + e t/ cos t sin t - 99
131 Example 2: y + 2y + 5y = cos t General solution y = C 1 e t cos 2t + C 2 e t sin 2t+ Transient solution: 1 5 cos t sin t y(t) = 2e t cos2t (C 1 = 2, C 2 = 0) Steady-State solution: Z(t) = 1 5 cos t sin t 100
132 Transient solution. Steady-State solution: 101
133 y = 2e t cos2t cos t sin t 102
Section 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationdx n a 1(x) dy
HIGHER ORDER DIFFERENTIAL EQUATIONS Theory of linear equations Initial-value and boundary-value problem nth-order initial value problem is Solve: a n (x) dn y dx n + a n 1(x) dn 1 y dx n 1 +... + a 1(x)
More information3.4.1 Distinct Real Roots
Math 334 3.4. CONSTANT COEFFICIENT EQUATIONS 34 Assume that P(x) = p (const.) and Q(x) = q (const.), so that we have y + py + qy = 0, or L[y] = 0 (where L := d2 dx 2 + p d + q). (3.7) dx Guess a solution
More information2. Second-order Linear Ordinary Differential Equations
Advanced Engineering Mathematics 2. Second-order Linear ODEs 1 2. Second-order Linear Ordinary Differential Equations 2.1 Homogeneous linear ODEs 2.2 Homogeneous linear ODEs with constant coefficients
More information2. Determine whether the following pair of functions are linearly dependent, or linearly independent:
Topics to be covered on the exam include: Recognizing, and verifying solutions to homogeneous second-order linear differential equations, and their corresponding Initial Value Problems Recognizing and
More informationConsider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity
1 Second Order Ordinary Differential Equations 1.1 The harmonic oscillator Consider an ideal pendulum as shown below. θ l Fr mg l θ is the angular acceleration θ is the angular velocity A point mass m
More informationChapter 3. Reading assignment: In this chapter we will cover Sections dx 1 + a 0(x)y(x) = g(x). (1)
Chapter 3 3 Introduction Reading assignment: In this chapter we will cover Sections 3.1 3.6. 3.1 Theory of Linear Equations Recall that an nth order Linear ODE is an equation that can be written in the
More informationAPPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai
APPLIED MATHEMATICS Part 1: Ordinary Differential Equations Contents 1 First Order Differential Equations 3 1.1 Basic Concepts and Ideas................... 4 1.2 Separable Differential Equations................
More informationMath 4B Notes. Written by Victoria Kala SH 6432u Office Hours: T 12:45 1:45pm Last updated 7/24/2016
Math 4B Notes Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: T 2:45 :45pm Last updated 7/24/206 Classification of Differential Equations The order of a differential equation is the
More informationOrdinary Differential Equations
Ordinary Differential Equations (MA102 Mathematics II) Shyamashree Upadhyay IIT Guwahati Shyamashree Upadhyay ( IIT Guwahati ) Ordinary Differential Equations 1 / 15 Method of Undetermined Coefficients
More informationMAT187H1F Lec0101 Burbulla
Spring 2017 Second Order Linear Homogeneous Differential Equation DE: A(x) d 2 y dx 2 + B(x)dy dx + C(x)y = 0 This equation is called second order because it includes the second derivative of y; it is
More information4.2 Homogeneous Linear Equations
4.2 Homogeneous Linear Equations Homogeneous Linear Equations with Constant Coefficients Consider the first-order linear differential equation with constant coefficients a 0 and b. If f(t) = 0 then this
More informationTheory of Higher-Order Linear Differential Equations
Chapter 6 Theory of Higher-Order Linear Differential Equations 6.1 Basic Theory A linear differential equation of order n has the form a n (x)y (n) (x) + a n 1 (x)y (n 1) (x) + + a 0 (x)y(x) = b(x), (6.1.1)
More information6 Second Order Linear Differential Equations
6 Second Order Linear Differential Equations A differential equation for an unknown function y = f(x) that depends on a variable x is any equation that ties together functions of x with y and its derivatives.
More informationMath Assignment 5
Math 2280 - Assignment 5 Dylan Zwick Fall 2013 Section 3.4-1, 5, 18, 21 Section 3.5-1, 11, 23, 28, 35, 47, 56 Section 3.6-1, 2, 9, 17, 24 1 Section 3.4 - Mechanical Vibrations 3.4.1 - Determine the period
More informationMath 211. Substitute Lecture. November 20, 2000
1 Math 211 Substitute Lecture November 20, 2000 2 Solutions to y + py + qy =0. Look for exponential solutions y(t) =e λt. Characteristic equation: λ 2 + pλ + q =0. Characteristic polynomial: λ 2 + pλ +
More informationSecond-Order Linear ODEs
C0.tex /4/011 16: 3 Page 13 Chap. Second-Order Linear ODEs Chapter presents different types of second-order ODEs and the specific techniques on how to solve them. The methods are systematic, but it requires
More informationMATHEMATICS FOR ENGINEERS & SCIENTISTS 23
MATHEMATICS FOR ENGINEERS & SCIENTISTS 3.5. Second order linear O.D.E.s: non-homogeneous case.. We ll now consider non-homogeneous second order linear O.D.E.s. These are of the form a + by + c rx) for
More informationDifferential Equations
Differential Equations A differential equation (DE) is an equation which involves an unknown function f (x) as well as some of its derivatives. To solve a differential equation means to find the unknown
More informationUNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH *
4.4 UNDETERMINED COEFFICIENTS SUPERPOSITION APPROACH 19 Discussion Problems 59. Two roots of a cubic auxiliary equation with real coeffi cients are m 1 1 and m i. What is the corresponding homogeneous
More informationChapter 2 Second-Order Linear ODEs
Chapter 2 Second-Order Linear ODEs Advanced Engineering Mathematics Wei-Ta Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2.1 Homogeneous Linear ODEs of Second Order 2 Homogeneous Linear ODEs
More informationIntroductory Differential Equations
Introductory Differential Equations Lecture Notes June 3, 208 Contents Introduction Terminology and Examples 2 Classification of Differential Equations 4 2 First Order ODEs 5 2 Separable ODEs 5 22 First
More informationChapter 2 Second Order Differential Equations
Chapter 2 Second Order Differential Equations Either mathematics is too big for the human mind or the human mind is more than a machine. - Kurt Gödel (1906-1978) 2.1 The Simple Harmonic Oscillator The
More informationBasic Theory of Linear Differential Equations
Basic Theory of Linear Differential Equations Picard-Lindelöf Existence-Uniqueness Vector nth Order Theorem Second Order Linear Theorem Higher Order Linear Theorem Homogeneous Structure Recipe for Constant-Coefficient
More informationSecond order linear equations
Second order linear equations Samy Tindel Purdue University Differential equations - MA 266 Taken from Elementary differential equations by Boyce and DiPrima Samy T. Second order equations Differential
More informationSecond-Order Linear ODEs
Chap. 2 Second-Order Linear ODEs Sec. 2.1 Homogeneous Linear ODEs of Second Order On pp. 45-46 we extend concepts defined in Chap. 1, notably solution and homogeneous and nonhomogeneous, to second-order
More informationMath 2Z03 - Tutorial # 6. Oct. 26th, 27th, 28th, 2015
Math 2Z03 - Tutorial # 6 Oct. 26th, 27th, 28th, 2015 Tutorial Info: Tutorial Website: http://ms.mcmaster.ca/ dedieula/2z03.html Office Hours: Mondays 3pm - 5pm (in the Math Help Centre) Tutorial #6: 3.4
More informationMath 266 Midterm Exam 2
Math 266 Midterm Exam 2 March 2st 26 Name: Ground Rules. Calculator is NOT allowed. 2. Show your work for every problem unless otherwise stated (partial credits are available). 3. You may use one 4-by-6
More informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More informationNonhomogeneous Linear Differential Equations with Constant Coefficients - (3.4) Method of Undetermined Coefficients
Nonhomogeneous Linear Differential Equations with Constant Coefficients - (3.4) Method of Undetermined Coefficients Consider an nth-order nonhomogeneous linear differential equation with constant coefficients:
More information17.2 Nonhomogeneous Linear Equations. 27 September 2007
17.2 Nonhomogeneous Linear Equations 27 September 2007 Nonhomogeneous Linear Equations The differential equation to be studied is of the form ay (x) + by (x) + cy(x) = G(x) (1) where a 0, b, c are given
More informationMath 3313: Differential Equations Second-order ordinary differential equations
Math 3313: Differential Equations Second-order ordinary differential equations Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX Outline Mass-spring & Newton s 2nd law Properties
More informationMath 2142 Homework 5 Part 1 Solutions
Math 2142 Homework 5 Part 1 Solutions Problem 1. For the following homogeneous second order differential equations, give the general solution and the particular solution satisfying the given initial conditions.
More informationMA 266 Review Topics - Exam # 2 (updated)
MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential
More informationA( x) B( x) C( x) y( x) 0, A( x) 0
3.1 Lexicon Revisited The nonhomogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x) F( x), A( x) dx dx The homogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x), A( x) dx
More informationMATH 23 Exam 2 Review Solutions
MATH 23 Exam 2 Review Solutions Problem 1. Use the method of reduction of order to find a second solution of the given differential equation x 2 y (x 0.1875)y = 0, x > 0, y 1 (x) = x 1/4 e 2 x Solution
More informationSection 3.7: Mechanical and Electrical Vibrations
Section 3.7: Mechanical and Electrical Vibrations Second order linear equations with constant coefficients serve as mathematical models for mechanical and electrical oscillations. For example, the motion
More informationENGI Second Order Linear ODEs Page Second Order Linear Ordinary Differential Equations
ENGI 344 - Second Order Linear ODEs age -01. Second Order Linear Ordinary Differential Equations The general second order linear ordinary differential equation is of the form d y dy x Q x y Rx dx dx Of
More informationFourier transforms. c n e inπx. f (x) = Write same thing in an equivalent form, using n = 1, f (x) = l π
Fourier transforms We can imagine our periodic function having periodicity taken to the limits ± In this case, the function f (x) is not necessarily periodic, but we can still use Fourier transforms (related
More informationNotes on the Periodically Forced Harmonic Oscillator
Notes on the Periodically orced Harmonic Oscillator Warren Weckesser Math 38 - Differential Equations 1 The Periodically orced Harmonic Oscillator. By periodically forced harmonic oscillator, we mean the
More informationODE. Philippe Rukimbira. Department of Mathematics Florida International University PR (FIU) MAP / 92
ODE Philippe Rukimbira Department of Mathematics Florida International University PR (FIU) MAP 2302 1 / 92 4.4 The method of Variation of parameters 1. Second order differential equations (Normalized,
More informationApplications of Linear Higher-Order DEs
Week #4 : Applications of Linear Higher-Order DEs Goals: Solving Homogeneous DEs with Constant Coefficients - Second and Higher Order Applications - Damped Spring/Mass system Applications - Pendulum 1
More informationAtoms An atom is a term with coefficient 1 obtained by taking the real and imaginary parts of x j e ax+icx, j = 0, 1, 2,...,
Atoms An atom is a term with coefficient 1 obtained by taking the real and imaginary parts of x j e ax+icx, j = 0, 1, 2,..., where a and c represent real numbers and c 0. Details and Remarks The definition
More informationM A : Ordinary Differential Equations
M A 2 0 5 1: Ordinary Differential Equations Essential Class Notes & Graphics D 19 * 2018-2019 Sections D07 D11 & D14 1 1. INTRODUCTION CLASS 1 ODE: Course s Overarching Functions An introduction to the
More informationSECOND-ORDER DIFFERENTIAL EQUATIONS
Chapter 16 SECOND-ORDER DIFFERENTIAL EQUATIONS OVERVIEW In this chapter we extend our study of differential euations to those of second der. Second-der differential euations arise in many applications
More information4.9 Free Mechanical Vibrations
4.9 Free Mechanical Vibrations Spring-Mass Oscillator When the spring is not stretched and the mass m is at rest, the system is at equilibrium. Forces Acting in the System When the mass m is displaced
More information5 Linear Dierential Equations
Dierential Equations (part 2): Linear Dierential Equations (by Evan Dummit, 2016, v. 2.01) Contents 5 Linear Dierential Equations 1 5.1 General Linear Dierential Equations...................................
More informationSecond Order Linear ODEs, Part II
Craig J. Sutton craig.j.sutton@dartmouth.edu Department of Mathematics Dartmouth College Math 23 Differential Equations Winter 2013 Outline Non-homogeneous Linear Equations 1 Non-homogeneous Linear Equations
More informationMB4018 Differential equations
MB4018 Differential equations Part II http://www.staff.ul.ie/natalia/mb4018.html Prof. Natalia Kopteva Spring 2015 MB4018 (Spring 2015) Differential equations Part II 0 / 69 Section 1 Second-Order Linear
More informationChapter 3: Second Order Equations
Exam 2 Review This review sheet contains this cover page (a checklist of topics from Chapters 3). Following by all the review material posted pertaining to chapter 3 (all combined into one file). Chapter
More informationA: Brief Review of Ordinary Differential Equations
A: Brief Review of Ordinary Differential Equations Because of Principle # 1 mentioned in the Opening Remarks section, you should review your notes from your ordinary differential equations (odes) course
More informationThe Harmonic Oscillator
The Harmonic Oscillator Math 4: Ordinary Differential Equations Chris Meyer May 3, 008 Introduction The harmonic oscillator is a common model used in physics because of the wide range of problems it can
More informationMath K (24564) - Lectures 02
Math 39100 K (24564) - Lectures 02 Ethan Akin Office: NAC 6/287 Phone: 650-5136 Email: ethanakin@earthlink.net Spring, 2018 Contents Second Order Linear Equations, B & D Chapter 4 Second Order Linear Homogeneous
More informationB Ordinary Differential Equations Review
B Ordinary Differential Equations Review The profound study of nature is the most fertile source of mathematical discoveries. - Joseph Fourier (1768-1830) B.1 First Order Differential Equations Before
More informationAdvanced Eng. Mathematics
Koya University Faculty of Engineering Petroleum Engineering Department Advanced Eng. Mathematics Lecture 6 Prepared by: Haval Hawez E-mail: haval.hawez@koyauniversity.org 1 Second Order Linear Ordinary
More informationOrdinary Differential Equations (ODEs)
Chapter 13 Ordinary Differential Equations (ODEs) We briefly review how to solve some of the most standard ODEs. 13.1 First Order Equations 13.1.1 Separable Equations A first-order ordinary differential
More information0.1 Problems to solve
0.1 Problems to solve Homework Set No. NEEP 547 Due September 0, 013 DLH Nonlinear Eqs. reducible to first order: 1. 5pts) Find the general solution to the differential equation: y = [ 1 + y ) ] 3/. 5pts)
More informationApplied Differential Equation. October 22, 2012
Applied Differential Equation October 22, 22 Contents 3 Second Order Linear Equations 2 3. Second Order linear homogeneous equations with constant coefficients.......... 4 3.2 Solutions of Linear Homogeneous
More informationHomework #6 Solutions
Problems Section.1: 6, 4, 40, 46 Section.:, 8, 10, 14, 18, 4, 0 Homework #6 Solutions.1.6. Determine whether the functions f (x) = cos x + sin x and g(x) = cos x sin x are linearly dependent or linearly
More informationIV Higher Order Linear ODEs
IV Higher Order Linear ODEs Boyce & DiPrima, Chapter 4 H.J.Eberl - MATH*2170 0 IV Higher Order Linear ODEs IV.1 General remarks Boyce & DiPrima, Section 4.1 H.J.Eberl - MATH*2170 1 Problem formulation
More informationMATH 1231 MATHEMATICS 1B Calculus Section 2: - ODEs.
MATH 1231 MATHEMATICS 1B 2007. For use in Dr Chris Tisdell s lectures: Tues 11 + Thur 10 in KBT Calculus Section 2: - ODEs. 1. Motivation 2. What you should already know 3. Types and orders of ODEs 4.
More information) sm wl t. _.!!... e -pt sinh y t. Vo + mx" + cx' + kx = 0 (26) has a unique tions x(o) solution for t ;?; 0 satisfying given initial condi
1 48 Chapter 2 Linear Equations of Higher Order 28. (Overdamped) If Xo = 0, deduce from Problem 27 that x(t) Vo = e -pt sinh y t. Y 29. (Overdamped) Prove that in this case the mass can pass through its
More informationMA Ordinary Differential Equations
MA 108 - Ordinary Differential Equations Santanu Dey Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 76 dey@math.iitb.ac.in March 26, 2014 Outline of the lecture Method
More informationLinear second-order differential equations with constant coefficients and nonzero right-hand side
Linear second-order differential equations with constant coefficients and nonzero right-hand side We return to the damped, driven simple harmonic oscillator d 2 y dy + 2b dt2 dt + ω2 0y = F sin ωt We note
More information17.8 Nonhomogeneous Linear Equations We now consider the problem of solving the nonhomogeneous second-order differential
ADAMS: Calculus: a Complete Course, 4th Edition. Chapter 17 page 1016 colour black August 15, 2002 1016 CHAPTER 17 Ordinary Differential Equations 17.8 Nonhomogeneous Linear Equations We now consider the
More informationThursday, August 4, 2011
Chapter 16 Thursday, August 4, 2011 16.1 Springs in Motion: Hooke s Law and the Second-Order ODE We have seen alrealdy that differential equations are powerful tools for understanding mechanics and electro-magnetism.
More informationM A : Ordinary Differential Equations
M A 2 0 5 1: Ordinary Differential Equations Essential Class Notes & Graphics C 17 * Sections C11-C18, C20 2016-2017 1 Required Background 1. INTRODUCTION CLASS 1 The definition of the derivative, Derivative
More informationForced Mechanical Vibrations
Forced Mechanical Vibrations Today we use methods for solving nonhomogeneous second order linear differential equations to study the behavior of mechanical systems.. Forcing: Transient and Steady State
More informationSolution of Constant Coefficients ODE
Solution of Constant Coefficients ODE Department of Mathematics IIT Guwahati Thus, k r k (erx ) r=r1 = x k e r 1x will be a solution to L(y) = 0 for k = 0, 1,..., m 1. So, m distinct solutions are e r
More informationUnforced Mechanical Vibrations
Unforced Mechanical Vibrations Today we begin to consider applications of second order ordinary differential equations. 1. Spring-Mass Systems 2. Unforced Systems: Damped Motion 1 Spring-Mass Systems We
More informationMATH 251 Examination I February 23, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination I February 23, 2017 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. Show all you your work! In order to obtain full credit for partial credit
More informationLinear Second Order ODEs
Chapter 3 Linear Second Order ODEs In this chapter we study ODEs of the form (3.1) y + p(t)y + q(t)y = f(t), where p, q, and f are given functions. Since there are two derivatives, we might expect that
More informationMa 221. The material below was covered during the lecture given last Wed. (1/29/14). Homogeneous Linear Equations with Constant Coefficients
Ma 1 The material below was covered during the lecture given last Wed. (1/9/1). Homogeneous Linear Equations with Constant Coefficients We shall now discuss the problem of solving the homogeneous equation
More informationSection 11.1 What is a Differential Equation?
1 Section 11.1 What is a Differential Equation? Example 1 Suppose a ball is dropped from the top of a building of height 50 meters. Let h(t) denote the height of the ball after t seconds, then it is known
More informationApplications of Second-Order Differential Equations
Applications of Second-Order Differential Equations ymy/013 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition
More informationLECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS
LECTURE 14: REGULAR SINGULAR POINTS, EULER EQUATIONS 1. Regular Singular Points During the past few lectures, we have been focusing on second order linear ODEs of the form y + p(x)y + q(x)y = g(x). Particularly,
More informationBackground ODEs (2A) Young Won Lim 3/7/15
Background ODEs (2A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any
More informationThe particular integral :
Second order linear equation with constant coefficients The particular integral : d f df Lf = a + a + af = h( x) Solutions with combinations of driving functions d f df Lf = a + a + af = h( x) + h( x)
More informationLecture Notes on. Differential Equations. Emre Sermutlu
Lecture Notes on Differential Equations Emre Sermutlu ISBN: Copyright Notice: To my wife Nurten and my daughters İlayda and Alara Contents Preface ix 1 First Order ODE 1 1.1 Definitions.............................
More informationSECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS A second-order linear differential equation has the form 1 Px d y dx Qx dx Rxy Gx where P, Q, R, and G are continuous functions. Equations of this type arise
More informationDiff. Eq. App.( ) Midterm 1 Solutions
Diff. Eq. App.(110.302) Midterm 1 Solutions Johns Hopkins University February 28, 2011 Problem 1.[3 15 = 45 points] Solve the following differential equations. (Hint: Identify the types of the equations
More informationDifferential Equations and Linear Algebra Lecture Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS
Differential Equations and Linear Algebra Lecture Notes Simon J.A. Malham Department of Mathematics, Heriot-Watt University, Edinburgh EH4 4AS Contents Chapter. Linear second order ODEs 5.. Newton s second
More informationODE Homework Solutions of Linear Homogeneous Equations; the Wronskian
ODE Homework 3 3.. Solutions of Linear Homogeneous Equations; the Wronskian 1. Verify that the functions y 1 (t = e t and y (t = te t are solutions of the differential equation y y + y = 0 Do they constitute
More informationNATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH421 COURSE TITLE: ORDINARY DIFFERENTIAL EQUATIONS
MTH 421 NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH421 COURSE TITLE: ORDINARY DIFFERENTIAL EQUATIONS MTH 421 ORDINARY DIFFERENTIAL EQUATIONS COURSE WRITER Prof.
More informationAPPM 2360: Midterm 3 July 12, 2013.
APPM 2360: Midterm 3 July 12, 2013. ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your instructor s name, (3) your recitation section number and (4) a grading table. Text books, class notes,
More informationForm A. 1. Which of the following is a second-order, linear, homogenous differential equation? 2
Form A Math 4 Common Part of Final Exam December 6, 996 INSTRUCTIONS: Please enter your NAME, ID NUMBER, FORM designation, and INDEX NUMBER on your op scan sheet. The index number should be written in
More information1+t 2 (l) y = 2xy 3 (m) x = 2tx + 1 (n) x = 2tx + t (o) y = 1 + y (p) y = ty (q) y =
DIFFERENTIAL EQUATIONS. Solved exercises.. Find the set of all solutions of the following first order differential equations: (a) x = t (b) y = xy (c) x = x (d) x = (e) x = t (f) x = x t (g) x = x log
More informationLecture 1: Review of methods to solve Ordinary Differential Equations
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce Not to be copied, used, or revised without explicit written permission from the copyright owner 1 Lecture 1: Review of methods
More informationA Brief Review of Elementary Ordinary Differential Equations
A A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on
More informationUndetermined Coefficents, Resonance, Applications
Undetermined Coefficents, Resonance, Applications An Undetermined Coefficients Illustration Phase-amplitude conversion I Phase-amplitude conversion II Cafe door Pet door Cafe Door Model Pet Door Model
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More informationVibrations and Waves MP205, Assignment 4 Solutions
Vibrations and Waves MP205, Assignment Solutions 1. Verify that x = Ae αt cos ωt is a possible solution of the equation and find α and ω in terms of γ and ω 0. [20] dt 2 + γ dx dt + ω2 0x = 0, Given x
More informationThe most up-to-date version of this collection of homework exercises can always be found at bob/math365/mmm.pdf.
Millersville University Department of Mathematics MATH 365 Ordinary Differential Equations January 23, 212 The most up-to-date version of this collection of homework exercises can always be found at http://banach.millersville.edu/
More information4r 2 12r + 9 = 0. r = 24 ± y = e 3x. y = xe 3x. r 2 6r + 25 = 0. y(0) = c 1 = 3 y (0) = 3c 1 + 4c 2 = c 2 = 1
Mathematics MATB44, Assignment 2 Solutions to Selected Problems Question. Solve 4y 2y + 9y = 0 Soln: The characteristic equation is The solutions are (repeated root) So the solutions are and Question 2
More informationMcGill University Math 325A: Differential Equations LECTURE 12: SOLUTIONS FOR EQUATIONS WITH CONSTANTS COEFFICIENTS (II)
McGill University Math 325A: Differential Equations LECTURE 12: SOLUTIONS FOR EQUATIONS WITH CONSTANTS COEFFICIENTS (II) HIGHER ORDER DIFFERENTIAL EQUATIONS (IV) 1 Introduction (Text: pp. 338-367, Chap.
More informationEx. 1. Find the general solution for each of the following differential equations:
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1.
More information1. (10 points) Find the general solution to the following second-order differential equation:
Math 307A, Winter 014 Midterm Solutions Page 1 of 8 1. (10 points) Find the general solution to the following second-order differential equation: 4y 1y + 9y = 9t. To find the general solution to this nonhomogeneous
More informationLinear DifferentiaL Equation
Linear DifferentiaL Equation Massoud Malek The set F of all complex-valued functions is known to be a vector space of infinite dimension. Solutions to any linear differential equations, form a subspace
More information