CHAPTER 1 FIRST-ORDER DIFFERENTIAL EQUATIONS



1. FIRST-ORDER DIFFERENTIAL EQUATIONS

1.1 Preliminary Concepts

1. General and particular solutions: For [pic], any equation involving a first derivative; [pic] such that [pic].

Example: [pic]

[pic]

[pic]

2. Implicitly defined solutions

Example: [pic]

3. Integral curves: a graph of a solution

4. The initial value problem: [pic], initial condition: [pic]

Example: [pic]

5. Direction fields: [pic]

1.2 Separable Equations

1. Separable differential equation: [pic]

Example: [pic]

RC circuits: Charging: [pic]

Discharging: [pic]

1.3 Linear Differential Equations: [pic], integrating factor: [pic]

Example: [pic]. [pic], [pic].

1.4 Exact Differential Equations

1. Potential function: For [pic], we can find a [pic] such that [pic] and [pic]; [pic] is the potential function; [pic] is exact.

2. Exact differential equation: a potential function exists; general solution: [pic].

Example: [pic].

3. Theorem: Test for exactness: [pic]

Example: ,[pic]. [pic].

1.5 Integrating Factors

1. Integrating factor: [pic] such that [pic] is exact.

Example: [pic].

2. How to find integrating factor: [pic]

Example: [pic].

3. Separable equations and integrating factors: [pic]

4. Linear equations and integrating factors: [pic]

1.6 Homogeneous and Bernoulli Equations

1. Homogeneous differential equation: [pic]; let [pic] separable.

Example: [pic].

2. Bernoulli equation: [pic]; [pic] linear; [pic] separable; otherwise, let [pic]linear

Example: [pic].

2. SECOND-ORDER DIFFERENTIAL EQUTIONS

2.1 Preliminary Concepts

1. [pic], an equation that contains a second derivative, but no higher derivative.

2. Linear second-order differential equations: [pic].

2.2 Theory of Solutions

1. The initial value problem: [pic]; [pic], [pic].

Example: [pic].

2. The homogeneous linear ODEs of 2nd order: [pic].

3. Theorem: Let [pic] and [pic] be solutions of [pic] on an interval I. Then any linear combination of these solutions, i.e., [pic], is also a solution.

4. Linear dependence: Two functions f and g are linearly dependent on an open interval I if, for some constant c, either [pic] for all x in I, or [pic] for all x in I. Linear independence: If f and g are not linearly dependent on I.

Example: [pic].

5. Wronskian Test: Let [pic] and [pic] be solutions of [pic] on an open interval I. Then, (1) Either [pic] for all x in I, or [pic] for all x in I. (2) [pic] and [pic] are linearly independent on I if and only if [pic] on I, where [pic].

Example: [pic].

6. Theorem: Let [pic] and [pic] be linearly independent solutions of [pic] on an open interval I. Then, every solution of this differential equation on I is a linear combination of [pic] and [pic].

7. Definition: Let [pic] and [pic] be solutions of [pic] on an open interval I. (1) [pic] and [pic] form a fundamental set (or a basis) of solutions on I if [pic] and [pic] are linearly independent on I. (2) When [pic] and [pic] form a fundamental set of solutions, we call [pic] , with [pic] and [pic] arbitrary constants, the general solution of the differential equation on I.

8. The nonhomogeneous equations: [pic].

9. Theorem: Let [pic] and [pic] be a fundamental set of solutions of [pic] on an open interval I. Let [pic] be any solution of [pic] on I. Then, for any solution [pic] of [pic], there exist numbers [pic] and [pic] such that [pic].

2.3 Reduction of Order: Given [pic], if we know a first solution [pic], then a second solution can be the form [pic].

Example: [pic].

2.4 The Constant Coefficient Homogeneous Linear Equation: [pic], A and B are numbers.

1. Characteristic equation: [pic] obtained by substituting [pic] into [pic].

2. Case 1. [pic]: The general solution is [pic]; [pic], [pic].

Example: [pic].

3. Case 2. [pic]: The general solution is [pic]; [pic].

Example: [pic].

4. Case 3. [pic]: The general solution is [pic]; [pic], [pic].

Example: [pic].

5. An alternative general solution in the complex root case: [pic].

Maclaurin expansions: [pic], [pic], [pic].

Euler’ formula: [pic].

Example: [pic].

2.5 Euler’s (Euler-Cauchy) Equation: [pic], let (i) [pic] [pic] Characteristic equation: [pic], or (ii) let [pic], [pic], [pic] [pic] [pic].

1. Case 1. [pic]: The general solution is [pic]; [pic], [pic].

Example: [pic].

2. Case 2. [pic]: The general solution is [pic]; [pic].

Example: [pic].

3. Case 3. [pic]: The general solution is [pic]; [pic], [pic].

Example: [pic].

2.6 The Nonhomogeneous Equation: [pic], general solution [pic].

1. The method of variation of parameters: let [pic], then simultaneously solve [pic].

Example: [pic].

2. The method of undetermined coefficients: only applied while p(x) and q(x) are constant, i.e., [pic].

[pic]

Example: [pic].

-- Modification Rule: If a term in your choice for [pic] happens to be a solution of the homogeneous ODE, multiply your choice of [pic] by x (or by [pic] if this solution corresponds to a double root of the characteristic equation of the homogeneous ODE).

Example: [pic].

[pic]

3. The principle of superposition: [pic], [pic] is a solution of [pic], then [pic] is a solution.

Example: [pic].

3. HIGHER ORDER LINEAR ODES

3.1 Homogeneous Linear ODEs

1. [pic], a nth order ODE if the nth derivative [pic] of the unknown function [pic] is the highest occurring derivative.

2. Linear ODE: [pic].

3. Homogeneous linear ODE: [pic].

4. Theorem: Fundamental Theorem for the Homogeneous Linear ODE: For a homogeneous linear ODE, sums and constant multiples of solutions on some open interval I are again solutions on I. (This does not hold for a nonhomogeneous or nonlinear ODE!).

5. General solution: [pic], where [pic]is a basis (or fundamental system) of solutions on I; that is, these solutions are linearly independent on I.

6. Linear independence and dependence: n functions [pic] are called linearly independent on some interval I where they are defined if the equation [pic] on I implies that all [pic] are zero. These functions are called linearly dependent on I if this equation also holds on I for some [pic] not all zero.

Example: [pic]. Sol.: [pic].

7. Theorem: Let the homogeneous linear ODE have continuous coefficients [pic], [pic] on an open interval I. Then n solutions [pic] on I are linearly dependent on I if and only if their Wronskian is zero for some [pic] in I. Furthermore, if W is zero for [pic], then W is identically zero on I. Hence if there is an [pic] in I at which W is not zero, then [pic] are linearly independent on I, so that they form a basis of solutions of the homogeneous linear ODE on I.

Wronskian: [pic]

8. Initial value problem: An ODE with n initial conditions [pic], [pic], [pic].

3.2 Homogeneous Linear ODEs with Constant Coefficients

1. [pic]: Substituting [pic], we obtain the characteristic equation [pic].

(i) Distinct real roots: The general solution is [pic]

Example: [pic]. Sol.: [pic].

(ii) Simple complex roots: [pic], [pic], [pic].

Example: [pic]. Sol.: [pic].

(iii) Multiple real roots: If [pic] is a real root of order m, then m corresponding linearly independent solutions are: [pic], [pic], [pic], [pic].

Example: [pic]. Sol.: [pic].

(iv) Multiple complex roots: If [pic] are complex double roots, the corresponding linearly independent solutions are: [pic], [pic], [pic], [pic].

2. Convert the higher-order differential equation to a system of first-order equations.

Example: [pic].

3.3 Nonhomogeneous Linear ODEs

1. [pic], the general solution is of the form: [pic], where [pic] is the homogeneous solution and [pic] is a particular solution.

2. Method of undermined coefficients

Example: [pic]. Sol.: [pic].

3. Method of variation of parameters: [pic], where [pic], [pic].

Example: [pic]. Sol.: [pic].

4. LAPLACE TRANSFORM

4.1 Definition and Basic Properties: initial value problem [pic] algebra problem [pic] solution of the algebra problem [pic] solution of the initial value problem

1. Definition (Laplace Transform): The Laplace transform [pic], for all s such that this integral converges.

Examples: [pic] [pic] [pic], [pic]. [pic] [pic] [pic].

2. Table of Laplace transform of functions

[pic]

3. Theorem (Linearity of the Laplace transform): Suppose [pic] and [pic] are defined for [pic], and [pic] and [pic] are real numbers. Then [pic] for [pic].

4. Definition (Inverse Laplace transform): Given a function G, a function g such that [pic] is called an inverse Laplace transform of G. In this event, we write [pic].

5. Theorem (Lerch): Let f and g be continuous on [pic] and suppose that [pic]. Then [pic].

6. Theorem: If [pic] and [pic] and [pic] and [pic] are real numbers, then [pic].

4.2 Solution of Initial Value Problems Using the Laplace Transform

1. Theorem (Laplace transform of a derivative): Let f be continuous on [pic] and suppose [pic] is piecewise continuous on [pic] for every positive k. Suppose also that [pic] if [pic]. Then [pic].

2. Theorem (Laplace transform of a higher derivative): Suppose [pic] are continuous on [pic] and [pic] is piecewise continuous on [pic] for every positive k. Suppose also that [pic] for [pic]and for [pic]. Then [pic].

Examples: [pic] [pic] [pic].

[pic] [pic] [pic].

4.3 Shifting Theorems and the Heaviside Function

1. Theorem (First shifting theorem, or shifting in the s variable): Let [pic] for [pic]. Let a be any number. Then [pic] for [pic]. [pic].

Examples: [pic] [pic] [pic]. Find [pic] [pic] [pic].

2. Definition (Heaviside function): The Heaviside function (or unit step function) H is defined by [pic]. [pic] [pic]

[pic] [pic]

-- On-off effect: [pic], [pic]

[pic]

3. Definition (Pulse): A pulse is a function of the form [pic], in which [pic].

[pic] [pic]

4. Theorem (Second shifting theorem, or shifting in the t variable): Let [pic] for [pic]. Then [pic] for [pic]. [pic] [pic]

Examples: [pic] [pic] [pic]. [pic] [pic] [pic].

Compute [pic], where [pic] for [pic] and [pic] for [pic]. [pic] [pic].

Solve [pic], [pic], [pic]. [pic] [pic].

4.4 Convolution

1. Definition (Convolution): If f and g are defined on [pic], then the convolution [pic] of f with g is the function defined by [pic] for [pic].

2. Theorem (Convolution theorem): If [pic] is defined, then [pic].

3. Theorem: Let [pic] and [pic]. Then [pic].

Example: [pic] [pic] [pic].

Determine f such that [pic] [pic] [pic].

4. Theorem: If [pic] is defined, so is [pic], and [pic].

Example: Solve [pic] [pic] [pic].

4.5 Unit Impulses and the Dirac’s Delta Function

1. Dirac’s delta function: [pic], where [pic]; [pic]; [pic].

[pic]

2. Theorem (Filtering property): Let [pic] and let f be integrable on [pic] and continuous at a. Then [pic]

-- Let [pic] [pic] [pic] [pic] the definition of the Laplace transformation of the delta function.

Example: Solve [pic] [pic] [pic].

4.6 Laplace Transform Solution of Systems

Example: Solve the system: [pic] [pic] [pic]

4.7 Differential Equations with Polynomial Coefficients

1. Theorem: Let [pic] for [pic] and suppose that F is differentiable. Then [pic] for [pic].

2. Corollary: Let [pic] for [pic] and let n be a positive integer. Suppose F is n times differentiable. Then [pic] for [pic].

Example: [pic] [pic] [pic].

3. Theorem: Let f be piecewise continuous on [pic] for every positive number k and suppose there are numbers M and b such that [pic] for [pic]. Let [pic]. Then [pic].

Example: [pic] [pic] [pic].

5. SERIES SOLUTIONS

5.1 Power Series Solutions of Initial Value Problems

1. Definition (Analytic function): A function f is analytical at [pic] if [pic] has a power series representation in some open interval about [pic]: [pic] in some interval [pic].

Example: Taylor series: [pic], [pic].

Maclaurin series: [pic]i.e., [pic] in Taylor series.

[pic] at [pic].

2. Theorem: Let p and q be analytic at [pic]. Then the initial value problem [pic]; [pic] has a solution that is analytical at [pic].

Example: [pic]; [pic] [pic] [pic]

3. Theorem: Let p, q and f be analytic at [pic]. Then the initial value problem [pic]; [pic], [pic]has a unique solution that is also analytical at [pic].

Examples: [pic]; [pic], [pic] [pic] [pic].

[pic] [pic] [pic], [pic], [pic].

5.2 Power Series Solutions Using Recurrence Relations

1. Coefficients developed to be a recurrence relation

Example: [pic] at [pic] [pic] [pic], [pic], [pic], [pic], [pic].

2. Two-term recurrence relation

Example: [pic] at [pic] [pic] [pic], [pic], [pic], [pic], [pic].

5.3 Singular Points and the Method of Frobenius

1. Definition (Ordinary and singular points): [pic] is an ordinary point of equation [pic] if [pic] and [pic], [pic], and [pic] are analytic at [pic]. [pic] is a singular point of equation [pic] if [pic] is not an ordinary point.

Example: [pic] [pic] [pic], [pic] are singular points.

2. Definition (Regular and irregular singular points): [pic] is a regular singular point of [pic] if [pic] is a singular point, and the functions [pic] and [pic] are analytic at [pic]. A singular point that is not regular is said to be an irregular singular point.

Example: [pic] [pic] [pic] is an irregular singular point, [pic] is a regular singular points.

3. Frobenius series: [pic].

4. Theorem (Method of Frobenius): Suppose [pic]is a regular singular point of [pic]. Then there exists at least one Frobenius solution [pic] with [pic]. Further, if the Taylor expansions of [pic] and [pic] about [pic] converge in an open interval [pic], then this Frobenius series also converges in this interval, except perhaps at [pic] itself.

There will be an indicial equation used to determine the values of r.

Example: [pic] [pic] [pic]: [pic], [pic]; [pic]: [pic], [pic]

5.4 Second Solutions and Logarithm Factors

1. Theorem (A second solution in the method of Frobenius): Suppose 0 is a regular singular point of [pic]. Let [pic] and [pic] be roots of the indicial equation. If these are real, suppose [pic]. Then (a) If [pic] is not an integer, there are two linearly independent Frobenius solutions: [pic] and [pic], with [pic] and [pic]. These solutions are valid in some interval [pic] or [pic]. (b) If [pic], there is a Frobenius solution [pic] with [pic] as well as a second solution: [pic]. Further, [pic] and [pic] form a fundamental set of solutions on some interval [pic]. (c) If [pic] is a positive integer, then there is a Frobenius solutions: [pic]. In this case there is a second solution of the form [pic]. If [pic] this is a second Frobenius series solution; if not, the solution contains a logarithm term. In either event, [pic] and [pic] form a fundamental set on some interval [pic].

Examples: [pic].[pic] [pic]: [pic], [pic], [pic], [pic].

[pic][pic] [pic] or [pic].

6. FOURIER SERIES

6.1 The Fourier Series of a Function

1. [pic], [pic], [pic] exists.

2. Lemma 13.1: If n and m are nonnegative integers, i.[pic]; ii.[pic], if [pic]; iii.[pic], if [pic].

3. Definition 13.1: Let f be a Riemann integrable function on [pic], then Fourier series of f on [pic]: [pic]; Fourier coefficients of f on [pic]: [pic], [pic] for [pic].

4. Definition 13.2: Even and odd functions:

f is an even function on [pic] if [pic] for [pic]; f is an odd function on [pic] if [pic] for [pic]; [pic]; [pic]; [pic].

[pic] if f is odd on [pic]; [pic] if f is even on [pic].

6.2 Convergence of Fourier Series

1. Definition 13.3: Piecewise continuous function

f is piecewise continuous on [a, b] if

1. f is continuous on [a, b] except perhaps at finitely many points.

2. Both [pic] and [pic] exist and are finite.

3. If [pic] is in (a, b) and f is not continuous at [pic], then [pic] and [pic] exist and are finite.

2. Definition 13.4: Piecewise smooth function

f is piecewise smooth on [a, b] if f and[pic] are piecewise continuous on [a, b].

3. Theorem 13.1: Convergence of Fourier series

Let f is piecewise smooth on [pic]. Then for [pic], the Fourier series of f on [pic] converge to[pic].

4. Convergence at the endpoints

5. Definition 13.5: Right derivative [pic]

6. Definition 13.6: Left derivative [pic]

7. Theorem 13.2: Let f is piecewise smooth on [pic]. Then,

i. If [pic] and f has a left and right derivative at x, then the Fourier series of f on [pic] converge at x to[pic].

ii. If [pic] and [pic] exist, then at both L and [pic], the Fourier series of f on [pic] converge to[pic].

8. Partial sums of Fourier series

6.3 Fourier Cosine and Sine Series

1. The Fourier cosine series of a function

Let f be integrable on the half-interval [0, L]: [pic], [pic] is an even function and called even extension of f on [pic].

Fourier cosine series of f on [0, L]: [pic]; Fourier cosine coefficients of f on [0, L]: [pic].

2. Theorem 13.3: Convergence of Fourier cosine series

Let f is piecewise continuous on [pic]. Then,

i. If [pic] and f has a left and right derivative at x, then the Fourier cosine series of f on [pic] converges at x to[pic].

ii. If f has a right derivative at 0, then the Fourier cosine series of f on [pic] converges at 0 to [pic].

iii. If f has a left derivative at L, then the Fourier cosine series of f on [pic] converges at L to [pic].

3. The Fourier sine series of a function

Let f be integrable on the half-interval [0, L]: [pic], [pic] is an odd function and called odd extension of f on [pic].

Fourier sine series of f on [0, L]: [pic]; Fourier sine coefficients of f on [0, L]: [pic].

4. Theorem 13.4: Convergence of Fourier sine series

Let f is piecewise continuous on [pic]. Then,

i. If [pic] and f has a left and right derivative at x, then the Fourier sine series of f on [pic] converges at x to[pic].

ii. At 0 and L, the Fourier sine series of f on [pic] converges to 0.

6.4 Integration and Differentiation of Fourier Series

1. Theorem 13.5: Integration of Fourier series

Let f be piecewise continuous on [pic], with Fourier series [pic]. Then, for any x on [pic], [pic].

2. Theorem 13.6: Differentiation of Fourier series

Let f be continuous on [pic] and suppose [pic]. Let [pic] be piecewise continuous on [pic]. Then, [pic], and at each point in [pic] where [pic] exists, [pic].

3. Theorem 13.7: Bessel’s inequalities: i. The coefficients in the Fourier sine expansion of f on [pic] satisfy [pic]; ii. The coefficients in the Fourier cosine expansion of f on [pic] satisfy [pic]; iii. If f is integrable on [pic], then the Fourier coefficients of f on [pic] satisfy [pic]

4. Theorem 13.8: Uniform and absolute convergence of Fourier series: Let f be continuous on [pic] and let[pic] be piecewise continuous. Suppose [pic]. Then, the Fourier series of f on [pic] converges absolutely and uniformly to [pic] on [pic].

5. Theorem 13.9: Parseval’s theorem: Let f be continuous on [pic] and let[pic] be piecewise continuous. Suppose [pic]. Then, the Fourier coefficients of f on [pic] satisfy [pic]

6.5 The Phase Angle Form of a Fourier Series

1. Periodic, fundamental period

2. Definition 13.7: Phase angle form: Let f have fundamental period p. Then the phase angle form of the Fourier series of f is [pic], in which [pic], [pic], and [pic] for [pic].

Harmonic form, nth harmonic, harmonic amplitude, phase angle.

6.6 Complex Fourier Series and the Frequency Spectrum

1. Conjugate, magnitude, argument, polar form.

2. Definition 13.8: Complex Fourier series: Let f have fundamental period p. Let [pic]. Then the complex Fourier series of f is [pic], where [pic] for [pic]. The numbers [pic] are the complex Fourier coefficients of f.

3. Theorem 13.10: Let f be periodic with fundamental period p. Let f be piecewise smooth on [pic]. Then at each x the complex Fourier series converges to [pic].

4. Amplitude spectrum; frequency spectrum.

[pic]

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[pic]

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