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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.
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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 containyand 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 • General 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 • 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 • 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 • 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: