Chap 1 First-Order Differential Equations 王 俊 鑫 (Chun-Hsin Wang) 中華大學 資訊工程系 Fall 2002

Outline Basic Concepts Separable Differential Equations substitution Methods Exact Differential Equations Integrating Factors Linear Differential Equations Bernoulli Equations

Basic Concepts Differentiation

Basic Concepts Differentiation

Basic Concepts Integration

Basic Concepts Integration

Basic Concepts Integration

Basic Concepts ODE vs. PDE Dependent Variables vs. Independent Variables Order Linear vs. Nonlinear Solutions

Basic Concepts Ordinary Differential Equations An unknown function (dependent variable) y of one independent variable x

Basic Concepts Partial Differential Equations An unknown function (dependent variable) z of two or more independent variables (e.g. x and y)

Basic Concepts The order of a differential equation is the order of the highest derivative that appears in the equation. Order 2 Order 1 Order 2

Basic Concept The first-order differential equation contain only y’ and may contain y and given function of x. A solution of a given first-order differential equation (*) on some open interval a<x<b is a function y=h(x) that has a derivative y’=h(x) and satisfies (*) for all x in that interval. (*) or

Basic Concept Example : Verify the solution

Basic Concepts Explicit Solution Implicit Solution

Basic Concept General solution vs. Particular solution arbitrary constant c Particular solution choose a specific c

Basic Concept Singular solutions Def : A differential equation may sometimes have an additional solution that cannot be obtained from the general solution and is then called a singular solution. Example The general solution : y=cx-c2 A singular solution : y=x2/4

Basic Concepts General Solution Particular Solution for y(0)=2 (initial condition)

Basic Concept Def: A differential equation together with an initial condition is called an initial value problem

Separable Differential Equations Def: A first-order differential equation of the form is called a separable differential equation

Separable Differential Equations Example : Sol:

Separable Differential Equations Example : Sol:

Separable Differential Equations Example : Sol:

Separable Differential Equations Example : Sol:

Separable Differential Equations Substitution Method: A differential equation of the form can be transformed into a separable differential equation

Separable Differential Equations Substitution Method:

Separable Differential Equations Example : Sol:

Separable Differential Equations Exercise 1

Exact Differential Equations Def: A first-order differential equation of the form is said to be exact if

Exact Differential Equations Proof:

Exact Differential Equations Example : Sol:

Exact Differential Equations Sol:

Exact Differential Equations Sol:

Exact Differential Equations Example

Non-Exactness Example :

Integrating Factor Def: A first-order differential equation of the form is not exact, but it will be exact if multiplied by F(x, y) then F(x,y) is called an integrating factor of this equation

Exact Differential Equations How to find integrating factor Golden Rule

Exact Differential Equations Example : Sol:

Exact Differential Equations Sol:

Exact Differential Equations Example :

Exact Differential Equations Exercise 2

Linear Differential Equations Def: A first-order differential equation is said to be linear if it can be written If r(x) = 0, this equation is said to be homogeneous

Linear Differential Equations How to solve first-order linear homogeneous ODE ? Sol:

Linear Differential Equations Example : Sol:

Linear Differential Equations How to solve first-order linear nonhomogeneous ODE ? Sol:

Linear Differential Equations Sol:

Linear Differential Equations Example : Sol:

Linear Differential Equations Example :

Bernoulli, Jocob Bernoulli, Jocob 1654-1705

Linear Differential Equations Def: Bernoulli equations If a = 0, Bernoulli Eq. => First Order Linear Eq. If a <> 0, let u = y1-a

Linear Differential Equations Example : Sol:

Linear Differential Equations Exercise 3

Summary

Orthogonal Trajectories of Curves Angle of intersection of two curves is defined to be the angle between the tangents of the curves at the point of intersection How to use differential equations for finding curves that intersect given curves at right angles ?

How to find Orthogonal Trajectories 1st Step: find a differential equation for a given cure 2nd Step: the differential equation of the orthogonal trajectories to be found 3rd step: solve the differential equation as above ( in 2nd step)

Orthogonal Trajectories of Curves Example: given a curve y=cx2, where c is arbitrary. Find their orthogonal trajectories. Sol:

Existance and Uniqueness of Solution An initial value problem may have no solutions, precisely one solution, or more than one solution. Example No solutions Precisely one solutions More than one solutions

Existence and uniqueness theorems Problem of existence Under what conditions does an initial value problem have at least one solution ? Existence theorem, see page 53 Problem of uniqueness Under what conditions does that the problem have at most one solution ? Uniqueness theorem, see page54