Chap 3 Linear Differential Equations 王 俊 鑫(Chun-Hsin Wang) 中華大學 資訊工程系 Fall 2002

Outline Second-Order Homogeneous Linear Equations Second-Order Homogeneous Equations with Constant Coefficients Modeling: Mass-Spring Systems, Electric Circuits Euler-Cauchy Equation Wronskian Second-Order Nonhomogeneous Linear Equations Higher Order Linear Differential Equations

Outline 歐拉-柯西 微分方程 二階線性齊次 常微分方程 高階線性 常微分方程 二階線性 常微分方程 線性 常微分方程 二階 常係數 二階線性齊次 常微分方程 歐拉-柯西 微分方程 二階線性齊次 常微分方程 二階線性非齊次 常微分方程 高階線性 常微分方程 二階線性 常微分方程 線性 常微分方程 二階 常微分方程

Second-Order ODE General Form for Second-Order Linear ODE Implicit Form Explicit Form

Second-Order Homogeneous Linear Equations Second-Order Homogeneous Linear ODE p(x), q(x): coefficient functions Example

Examples of Nonlinear differential equations

A linear combination of Solutions for homogeneous linear equation Example:

Second-Order Homogeneous Linear Equations Linear Principle (Superposition Principle) y is called the linear combination of y1 and y2 If y1 and y2 are the solutions of y = c1y1+ c2y2 is also a solution (c1, c2 arbitrary constants)

Second-Order Homogeneous Linear Equations Proof: Note

Does the Linearity Principle hold for nonhomogeneous linear or nonlinear equations ? Example: A nonhomogeneous linear differential equation Example: A nonlinear differential equation

Initial Value Problem for Second-Order homogeneous linear equations a general solution will be of the form , a linear combination of two solutions involving two arbitrary constants c1 and c2 An initial value problem consists two initial conditions.

Initial Value Problem Example: Observation: Our solution would not have been general enough to satisfy the two initial conditions and solve the problem.

A General Solution of an Homogeneous Linear Equation Definition: A general solution of an equation on an open interval I is a solution with y1 and y2 not proportional solutions of the equation on I and c1 ,c2 arbitrary constants. The y1 and y2 are then called a basis (or fundamental system) of the equation on I A particular solution of the equation is obtained if we assign specific values to c1 ,c2

Linear Independent Two functions y1(x) and y2(x) are linear independent on an interval I where they are defined if Example

How to obtain a Bass if One Solution is Known ? Method of Reduction Order Given y1 Find y2

Second-Order Homogeneous Linear Equations Proof:

Second-Order Homogeneous Linear Equations Proof:

Second-Order Homogeneous Linear Equations Example 3-1: Sol:

Second-Order Homogeneous Linear Equations Exercise 3-1: Basic Verification and Find Particular Solution Basis Initial Condition Basis Initial Condition Basis Initial Condition

Exercise: Reduce of order if a solution is known.

Second-Order Homogeneous Equations with Constant Coefficients General Form of Second-Order Homogeneous Equations with Constant Coefficients whose coefficients a and b are constant.

Second-Order Homogeneous Equations with Constant Coefficients Sol: Characteristic Equation

Second-Order Homogeneous Equations with Constant Coefficients Case 1: 兩相異實根 Case 2: 重根 Case 3: 共軛虛根

Second-Order Homogeneous Equations with Constant Coefficients Example 3-2: Sol: Step 1: Find General Solution

Second-Order Homogeneous Equations with Constant Coefficients Step 2: Find Particular Solution

Second-Order Homogeneous Equations with Constant Coefficients Step 3: Plot Particular Solution MATLAB Code x=[0:0.01:2]; y=exp(x)+3*exp(-2*x); plot(x,y)

Case 2 Real Double Root = -a/2

Second-Order Homogeneous Equations with Constant Coefficients Example 3-3: Sol: Step 1: Find General Solution

Second-Order Homogeneous Equations with Constant Coefficients Step 2: Find Particular Solution

Second-Order Homogeneous Equations with Constant Coefficients Step 3: Plot Particular Solution MATLAB Code x=[0:0.01:2]; y=(3-5*x).*exp(2*x); plot(x,y)

Euler Formula Euler Formula Proof: Maclaurin Series

Euler Formula Proof:

Euler Formula 幾何 虛數 分析 自然數 負數

Complex Exponential Function

Case 3

Second-Order Homogeneous Equations with Constant Coefficients Example 3-4: Sol: Step 1: Find General Solution

Second-Order Homogeneous Equations with Constant Coefficients Step 2: Find Particular Solution

Second-Order Homogeneous Equations with Constant Coefficients Step 3: Plot Particular Solution MATLAB Code x=[0:0.1:30]; y=exp(-0.1*x).*sin(2*x); plot(x,y)

Second-Order Homogeneous Equations with Constant Coefficients Exercise 3-2: Find General Solution 兩相異實根 重根 共軛虛根

Modeling: Mass-Spring Systems

Modeling: Electric Circuits Capacitor (farads) Resistor (ohms) Inductor (heries)

Modeling

Modeling Overdamping

Modeling Critical Damping

Modeling Underdamping

Euler-Cauchy Equation The Auxiliary Equation

Euler-Cauchy Equation Case 1: Distinct Real Roots m1, m2 Example 3-5:

Euler-Cauchy Equation Case 2: Double Roots m=(1-a)/2

Euler-Cauchy Case 2 :Example

Euler-Cauchy Equation Case 3: Complex Roots m = a ± bi

Euler-Cauchy Case 3 :Example

Existence and Uniqueness Theory If p(x) and q(x) are continuous function on some open interval  and x0 is in , then the initial value problem consisting of (1) and (3) has a unique solution y(x) on the interval .

Wronskian A set of n functions y1(x), y2(x), …, yn(x), is said to be linearly dependent over an interval I if there exist n constants c1, c2, …, cn, not all zero, such that Otherwise the set of functions is said to be linearly independent

Wronskian A set of n functions y1(x), y2(x), …, yn(x), is linearly independent over an interval I if and only if the determinant (Wronski determinant, or Wronskian)

Wronskian Example 3-8: Sol:  cosx, sinx are linearly independent

Linear Dependence and Independence of Solution Suppose that (1) has continuous coefficients p(x) and q(x) on an open interval . Then two solutions y1 and y2 of (1) on  are linear dependent on  if and only if their Wronskian W is zero at some x0 in . Furthermore, if W=0 for x= x0, then W=0 on ; hence if there is an x1in  at which W is not zero, then y1 ,y2 are liner independent on  .

Illustration of Theorem 2 Example 1 Example 2

A General Solution of (1) includes All Solutions Theorem 3 (Existence of a general solution) If p(x) and q(x) are continuous on an open interval , then (1) has a general solution on . Theorem 4 (General solution) Suppose that (1) has continuous coefficients p(x) and q(x) on some open interval . Then every solution y=Y(x) of (1) is of the form where y1 , y2 form a basis of solutions of (1) on  and c1, c2 are suitable constants. Hence (1) does not have singular solutions (I.e., solutions not obtainable from a general solution)

Nonhomogeneous Equations Theorem (a) The difference of two solutions of (1) on some open interval  is a solution of (2) on  (b) The sum of a solution of (1) and a solution of (2) on  is a solution of (1) on 

A general solution of the nonhomogeneous equation (1) on some open interval  is a solution of the form where yh(x)=c1y1(x)+c2y2(x) is a general solution of the homogeneous equation (2) on  and yp(x) is any solution of (1) on  containing no arbitrary constants. A particular solution of (1) on  is a solution obtain from (3) by assigning specific values to the arbitrary constants c1 and c2 in yh(x).

Practical Conclusion To solve the nonohomegeneous equation (1) or an initial value problem for (1) , we have to solve the homogeneous equation (2) and find any particular solution yp of (1)

Initial value problem for a nonhomogeneous equation Example

Solution by Undetermined Coefficients Method of Undetermined Coefficients General Solution: y = yh + yp yh : Homogeneous Solution yp : Particular Solution

Solution by Undetermined Coefficients

Rules for the Method of Undetermined Coefficients Basic Rule Modification Rule Sum Rule

Solution by Undetermined Coefficients Example 3-9: Sol:

Example for Modification Rule Example 1: in the case of a simple root Example 2: in the case of a double root Example 3: sum rule.

Second-Order Non-homogeneous Linear Equations Method of Variation of Parameters Particular Solution: y1, y2 : Homogeneous Solutions W : Wronskian of y1 and y2

Second-Order Non-homogeneous Linear Equations Example 3-10: Sol:

Higher Order Linear Differential Equations Higher Order Homogeneous Linear ODE If y1, y2, …, yn are the solutions of y = c1y1+ c2y2 +… + cnyn will be the general solution

Higher Order Linear Differential Equations Higher Order Nonhomogeneous Linear ODE General Solution: y = yh + yp yh : Homogeneous Solution yp : Particular Solution