# Chap 1 First-Order Differential Equations

## Presentation on theme: "Chap 1 First-Order Differential Equations"— Presentation transcript:

Chap 1 First-Order Differential Equations

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

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