Advanced Engineering Mathematics 6th Edition, Concise Edition by Peter V. O’Neil

Ordinary Differential Equations Part 1: Ordinary Differential Equations Ch1: First-Order Differential Equations Ch2: Second-Order Differential Equations Ch3: The Laplace Transform Ch4: Series Solutions

Differential equation --- contains derivatives e.g., or or y : function of x, x : independent variable (b) Ordinary differential equation – involves only total derivatives Partial differential equation – involves partial derivatives What is

(c) The order of a differential equation e. g (c) The order of a differential equation e.g., ○ The solution of a differential equation is a function y(x) of independent variable x that may be defined on e.g., i) , solution: y = sin2x for ii) , solution: y = xlnx – x for x > 0

Ch. 1: First-Order Differential Equations 1.1. Preliminary Concepts ○ First-order differential equation: -- involves a first but no higher derivatives e.g., y：function of x x：independent variable : solution

1.1.1. General and Particular Solutions ○ General solution: arbitrary constant Substitute into (A) Particular solutions: k = 1, ; k = 2 , k = ,

1.1.2. Implicitly Defined Solutions ○ Explicit function: Implicit function: e.g., , ○ (Explicit solution) (Implicit solution)

1.1.3. Integral Curves ○ Example 1.1: General solution: -- Help to comprehend the behavior of solution ○ Example 1.1: General solution:

1.1.4. Initial Value Problems ○ initial condition Graphically, the particular integral curve passes through point ( ) The objective is to obtain a unique solution ○ Example 1.4: , initial condition General solution: ,

-- A set of line segments tangent to a curve 1.1.5. Direction Field -- A set of line segments tangent to a curve -- Give a rough outline of the shape of the curve ○ Giving , instead of solving for y, solving for

-- A set of line segments tangent to a curve 1.1.5. Direction Field -- A set of line segments tangent to a curve -- Give a rough outline of the shape of the curve ○ Giving , instead of solving for y, solving for 11

Slope: , General Solution: ○ Example 1.5: Slope: , General Solution: Figure 1.5. Direction field for y'= y² and integral curves through (0,1), (0,2), (0,3), - 1, (0, - 2), and (0, - 3).

1.2. Separable Equations

○ Example 1.7: , (the general solution) y = 0 is a solution, called a singular solution, it cannot be obtained from the general solution

1.3. Linear Differential Equations -------- (A) (1) Find integrating factor: (2) Multiply (A) by (3)

○ Example 1.14: i) Integrating factor ii) Multiply the equation by , iii) iv) Integrate

1.4. Exact Differential Equations can be written as ------- (A) If , s.t. and (implicitly define the solution) : potential function

○ Example 1.17: Let From

＊ We can start with , then as well From Solution: ＊ We can start with , then as well 19

c(y) is not independent of x No potential function ＊ Not every is exact e.g. From c(y) is not independent of x No potential function

Theorem 1.1: Exactness is exact iff (a) If is exact, then s.t. (b) If , show s.t.

Let : any point

○ Example : 1.5. Integrating Factors If : not exact But : exact is not exact 1.5. Integrating Factors If : not exact But : exact integrating factor

. ○ How to find : exact, Try as , or ○ Example 1.21: The equation is not exact

Consider Let Try (B) Integrate

(A) Let be the potential function The implicit solution: The explicit solution:

○ Example 1.22: Let Find integrating factor by

Try This cannot be solved for as a function of x (ii) Likewise, let

(iii) Try (B) Divide by

: independent Multiply (A) by Let Form

From Obtain the potential function The implicit solution: or

1.6. Homogeneous, Bernoulli, and Riccati Eqs. 1.6.1 Homogeneous Equation: ----- (A) ＊ A homogeneous equation is always transformed into a separable one by letting ↑=1 (A)

○ Example 1.25: Let (A)

1.6.2. Bernoulli Equation linear separable can be transformed into linear by

○ Example 1.27: Let (A) Multiply by (linear) Integrating factor:

1.6.3. Riccati Equation Let S(x) be a solution and let The Riccati equation is transformed into linear ○ Example 1.28: By inspection, is a solution of (A) Let

(linear) Integrating factor: Integrate and Solution:

Initial Value Problems: 1.8. Existence and Uniqueness for Solutions of Initial Value Problems Initial Value Problems: The problem may have no solution and may have multiple solutions ○ Example 1.30: The equation is separable and has solution The equation has no real solution. is not a solution because it does not satisfy the initial condition.

○ Example 1.31: The equation has solution This problem has multiple solutions i, Trivial solution: ii, Define Consider All satisfy the initial condition

○ Theorem 1.2： If f, : continuous in , then s.t. The initial value problem has a unique solution defined on The size of h depends on f and

the entire plane and hence on ○ Example 1.31: not continuous on (x, 0) ○ Example：The problem : both continuous on the entire plane and hence on s.t. the problem has a unique solution in Solve the problem , which is valid for , we can take 41

○ Theorem 1.3： : continuous on I The problem , has a unique solution defined Proof： From the general solution of the linear equation and the initial condition , the solution of the initial value problem is : continuous on I, the solution is defined

Homework 1 Chapter 1 Sec.1.1: 1, 2, 7, 12 Sec.1.2: 1, 2, 11 Sec.1.3: 1 Sec.1.4: 1 Sec.1.5: 1

1.5(1): Determine a test involving M and N to tell when has an integrating factor that is a function of y only. Ans: Let be an integrating factor such that is exact. Then,

must be independent of x. The test is then that must be independent of x. 45

Homework 2 Chapter 1 Sec.1.6: 15, 16, 20, 21 Sec.1.8: 1, 3, 5

Chapter 2: Second-Order Differential Equations 2.1. Preliminary Concepts ○ Second-order differential equation e.g., Solution: A function satisfies , (I : an interval)

○ Linear second-order differential equation Nonlinear: e.g., 2.2. Theory of Solution ○ Consider y contains two parameters c and d

The graph of Given the initial condition

Given another initial condition The graph of ◎ The initial value problem: ○ Theorem 2.1: : continuous on I, has a unique solution

○ Theorem 2.2: : solutions of Eq. (2.2) 2.2.1.Homogeous Equation ○ Theorem 2.2: : solutions of Eq. (2.2) solution of Eq. (2.2) : real numbers Proof:

※ Two solutions are linearly independent. Their linear combination provides an infinity of new solutions ○ Definition 2.1: f , g : linearly dependent If s.t. or ; otherwise f , g : linearly independent In other words, f and g are linearly dependent only if for

○ Wronskian test -- Test whether two solutions of a homogeneous differential equation are linearly independent Define: Wronskian of solutions to be the 2 by 2 determinant

○ Let If : linear dep., then or Assume

○ Theorem 2.3: 1) Either or 2) : linearly independent iff Proof (2): (i) (if : linear indep. (P), then (Q) if ( Q) , then : linear dep. ( P) ) : linear dep.

(ii) (if (P), then : linear indep. (Q) if : linear dep. ( Q), then ( P)) : linear dep., ※ Test at just one point of I to determine linear dependency of the solutions

。 Example 2.2: are solutions of : linearly independent

。 Example 2.3: Solve by a power series method The Wronskian of at nonzero x would be difficult to evaluate, but at x = 0 are linearly independent

○ Definition 2.2: ◎ Find all solutions 1. : linearly independent : fundamental set of solutions 2. : general solution : constant ○ Theorem 2.4: : linearly independent solutions on I Any solution is a linear combination of

Proof: Let be a solution. Show s.t. Let and Then, is the unique solution on I of the initial value problem

2. 2. 2. Nonhomogeneous Equation ○ Theorem 2 2.2.2. Nonhomogeneous Equation ○ Theorem 2.5: : linearly independent homogeneous solutions of : a nonhomogeneous solution of any solution has the form

Proof: Given , solutions : a homogenous solution of : linearly independent homogenous solutions (Theorem 2.4)

1. Find the general homogeneous solutions ○ Steps: 1. Find the general homogeneous solutions of 2. Find any nonhomogeneous solution of 3. The general solution of is 2.3. Reduction of Order -- A method for finding the second independent homogeneous solution when given the first one

○ Let Substituting into ( : a homogeneous solution ) Let (separable)

For symlicity, let c = 1, 。 Example 2.4: : a solution Let : independent solutions 。 Example 2.4: : a solution Let

Substituting into (A), For simplicity, take c = 1, d = 0 : independent The general solution:

2. 4. Constant Coefficient Homogeneous A, B : numbers ----- (2 2.4. Constant Coefficient Homogeneous A, B : numbers ----- (2.4) The derivative of is a constant (i.e., ) multiple of Constant multiples of derivatives of y , which has form , must sum to 0 for (2,4) ○ Let Substituting into (2,4), (characteristic equation)

i) Solutions : : linearly independent The general solution:

。 Example 2.6: Let , Then Substituting into (A), The characteristic equation: The general solution:

ii) By the reduction of order method, Let Substituting into (2.4)

Choose : linearly independent The general sol. : 。 Example 2 Choose : linearly independent The general sol.: 。 Example 2.7: Characteristic eq. : The repeated root: The general solution:

iii) Let The general sol.:

。 Example 2.8: Characteristic equation: Roots: The general solution: ○ Find the real-valued general solution 。 Euler’s formula:

Maclaurin expansions:

。 Eq. (2.5),

Find any two independent solutions Take The general sol.:

2.5. Euler’s Equation , A , B : constants -----(2.7) Transform (2.7) to a constant coefficient equation by letting

Substituting into Eq. (2. 7), i. e. , --------(2 Substituting into Eq. (2.7), i.e., --------(2.8) Steps: (1) Solve (2) Substitute (3) Obtain

Characteristic equation: Roots: General solution: 。 Example 2.11: ------(A) -------(B) (i) Let Substituting into (A) Characteristic equation: Roots: General solution:

○ Solutions of constant coefficient linear equation have the forms: Solutions of Euler’s equation have the forms:

2.6. Nonhomogeneous Linear Equation ------(2.9) The general solution: ◎ Two methods for finding (1) Variation of parameters -- Replace with in the general homogeneous solution Let Assume ------(2.10) Compute

Substituting into (2.9), -----------(2.11) Solve (2.10) and (2.11) for Likewise,

。 Example 2. 15: ------(A) i) General homogeneous solution : Let 。 Example 2.15: ------(A) i) General homogeneous solution : Let . Substitute into (A) The characteristic equation: Complex solutions: Real solutions: :independent

ii) Nonhomogeneous solution Let

iii) The general solution:

(2) Undetermined coefficients Apply to A, B: constants Guess the form of from that of R e.g. : a polynomial Try a polynomial for : an exponential for Try an exponential for

。 Example 2.19: ---(A) It’s derivatives can be multiples of or Try Compute Substituting into (A),

: linearly independent and The homogeneous solutions: The general solution:

。 Example 2. 20: ------(A) , try Substituting into (A), 。 Example 2.20: ------(A) , try Substituting into (A), * This is because the guessed contains a homogeneous solution Strategy: If a homogeneous solution appears in any term of , multiply this term by x. If the modified term still occurs in a homogeneous solution, multiply by x again

Try Substituting into (A),

○ Steps of undetermined coefficients: (1) Find homogeneous solutions (2) From R(x), guess the form of If a homogeneous solution appears in any term of , multiply this term by x. If the modified term still occurs in a homogeneous solution, multiply by x again (3) Substitute the resultant into and solve for its coefficients

○ Guess from Let : a given polynomial , : polynomials with unknown coefficients Guessed

2.6.3. Superposition Let be a solution of is a solution of (A)

。 Example 2.25: The general solution: where homogeneous solutions

Chapter 3: The Laplace Transform 3.1. Definition and Basic Properties 。 Objective of Laplace transform -- Convert differential into algebraic equations ○ Definition 3.1: Laplace transform s.t. converges s, t : independent variables ＊ Representation:

。Example 3.2: Consider

Not every function has a Laplace transform * Not every function has a Laplace transform. In general, can not converge 。Example 3.1:

○ Definition 3.2.: Piecewise continuity (PC) f is PC on if there are finite points s.t. and are finite i.e., f is continuous on [a, b] except at finite points, at each of which f has finite one-sided limits

If f is PC on [0, k], then so is and exists

◎ Theorem 3.2: Existence of f is PC on If Proof:

* Theorem 3.2 is a sufficient but not a necessary condition.

＊ There may be different functions whose Laplace transforms are the same e.g., and have the same Laplace transform ○ Theorem 3.3: Lerch’s Theorem ＊ Table 3.1 lists Laplace transforms of functions

○ Theorem 3. 1: Laplace transform is linear Proof: ○ Definition 3. 3: ○ Theorem 3.1: Laplace transform is linear Proof: ○ Definition 3.3:. Inverse Laplace transform e.g., ＊ Inverse Laplace transform is linear

3.2 Solution of Initial Value Problems Using Laplace Transform ○ Theorem 3.5: Laplace transform of f: continuous on : PC on [0, k] Then, ------(3.1)

Proof: Let

○ Theorem 3.6: Laplace transform of : PC on [0, k] for s > 0, j = 1,2 … , n-1

。 Example 3.3: From Table 3.1, entries (5) and (8)

○ Laplace Transform of Integral From Eq. (3.1),

3.3. Shifting Theorems and Heaviside Function 3.3.1.The First Shifting Theorem ◎ Theorem 3.7: ○ Example 3.6: Given

○ Example 3.8:

3.3.2. Heaviside Function and Pulses ○ f has a jump discontinuity at a, if exist and are finite but unequal ○ Definition 3.4: Heaviside function 。 Shifting

。 Laplace transform of heaviside function

3.3.3 The Second Shifting Theorem Proof:

○ Example 3.11: Rewrite

◎ The inverse version of the second shifting theorem ○ Example 3.13: where rewritten as

3.4. Convolution

◎ Theorem 3.9: Convolution theorem Proof:

◎ Theorem 3.10: ○ Exmaple 3.18 ◎ Theorem 3.11: Proof :

○ Example 3.19:

3.5 Impulses and Dirac Delta Function ○ Definition 3.5: Pulse ○ Impulse: ○ Dirac delta function: A pulse of infinite magnitude over an infinitely short duration

○ Laplace transform of the delta function ◎ Filtering (Sampling) ○ Theorem 3.12: f : integrable and continuous at a

Proof:

by Hospital’s rule ○ Example 3.20:

3.6 Laplace Transform Solution of Systems ○ Example 3.22 Laplace transform Solve for

Partial fractions decomposition Inverse Laplace transform

3.7. Differential Equations with Polynomial Coefficient ◎ Theorem 3.13: Proof: ○ Corollary 3.1:

○ Example 3.25: Laplace transform

Find the integrating factor, Multiply (B) by the integrating factor

Inverse Laplace transform

○ Apply Laplace transform to algebraic expression for Y Differential equation for Y

◎ Theorem 3.14: PC on [0, k],

○ Example 3.26: Laplace transform ------(A) ------(B)

Finding an integrating factor, Multiply (B) by ,

In order to have

Formulas: ○ Laplace Transform: ○ Laplace Transform of Derivatives: ○ Laplace Transform of Integral:

○Shifting Theorems: ○ Convolution: Convolution Theorem: ○

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