Differential Equations Notes Chapters 1-3

7/29/2019 Differential Equations Notes Chapters 1-3

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DifferentialEquationsChapters13

1.1: Definitions and Terminology

Differential Equation: An Equation containing the derivatives of one or more independent

variables with respect to(wrt) one or more independent variables

Classifications By Type:

o Ordinary Differential Equations (ODE): An equation that contains only ordinaryderivatives of one or more dependent variables wrt a single independent variable

ex: 5 ,

6 0,

2

o Partial Differential Equations (ODE): An equation that contains partialderivatives of one or more dependent variables wrt two or more independent

variables

ex:

0 Classification by order:

o The order of a differential equation is the order of the highest derivative in theequation.

ex: 0, 1,

2 Classification by linearity:

o An nth order differential equation is linear if the following conditions are true: The dependent variable y & its derivatives (y ,y) are of the 1st

degree

The coefficients of a0, a1,a2 Of y, y,ydepend on at most x (theywill not be depend on y or any other variable)

Linear Equation takes form:

. .

Solution: Any function , defined on an interval I and possessing at least n derivatives thatare continuous on I, which when substituted into an nth-order ODE reduces the equation to

an identity, is said to be a solution of the equation on the interval.

Interval of existence: The domain of a solutiono Explicit vs. Implicit solutions

An Explicit Solutionis a solution where the dependent variable is expressedsolely in terms of the independent variable is explicit

A relation G(x, y) = 0 is said to be an ImplicitSolutionof an ordinarydifferential equation (4) on an interval I, provided that there exists at least one

function that satisfies the relation as well as the differential equation on I.

1.2: Initial Value Problems


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An Initial Value Problem(IVP) is an ODE that is paired with conditions imposed on unknown

function y(x)

First order IVPo Solve

,

o subject to Second order IVP

o Solve , , o subject to and

Existence and Uniquenesso Existence: does the ODE , possess any solutions? Di any of them pas

through ,o Uniqueness: When Can we be certain that there is precisely one solution passing

through ,o Theorem: Existence of a unique solution

Let R be a rectangular region in the xy-plane defined by axb;cydthatcontains,initsinterior.If, arecontinuousonRthenthereexistsomeintervalI0containedina,b,andauniquefunctionyxdefinedonI0thatisasolutiontotheivp

o Inotherwords,UniqueSolutionsExistWhere , is continuous

is contiuous2.2: Separable Variables

A first-order differential equation of the form is said to be separable or to have

separable variables.

To Solve a Separable variable equation:1) Group all y- terms on one side of the equation, all x-terms on the other side.

dy& dx must both be in the numerator2)

Integratebothsides

3) Ifanexplicitsolutionisneeded;solvefory


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Ex: Quiz 2:

2

3 2

3

2.3: Linear Equations

A first order DFEq in the form is a linear equation for the dependentvariable y

If g(x)= 0 then the function is homogeneous If g(x) 0 then the function is non-homogeneous

o Standard form found by dividing above equation by giving:o P where P

To Solve First Order ODE by integrating factor (I.F):

1)

Find the standard form by deviding by the coefficient of dy/dx2) Use P(x) from standard form to identify the integrating factor for the ODE

. 3) Multiply all terms of standard form by I.F. Left hand side always reduces to the

derivative of your integrating times y so the result is

4) Integrate both sides and solve for y

Any term whose contribution to the solution becomes negligible( approaches zero) asx is called a transient term

Any point which makes 0 causes a discontinuity on P(x) is called a singularpoint of the ODE

2.4: Exact Differential and Exact Equation

A differential expression, , is an exact differentialin a region R of thexy-plane if it corresponds to the differential of some function f(x,y) defined in R. A first order

ODE of the form Mx, ydxNx, ydy 0 is an exact equation if the Left Hand Side is anexact differential.


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2.5: Solutions by Substitution

HOMOGENEOUS EQUATIONS a functionf thatpossesses the property

, , ) for some real number, thenfis said to be a homogeneousfunctionof degree

Mx, ydxNx, ydy 0issaidtobehomogeneous*ifbothcoefficientfunctionsMandNarehomogeneousequationsofthesamedegree

To solve homogeneous equations:1 Substitute or 2 Simplify separable equation3 Solve using separation of variables

Ex: HW set 3 Mx, y y yx; Nx, y x

, ;,

Botharehomogeneousequationswith2 , Substituteyanddyforeq.above

0 0

2 0 2 1

2

1

2

1

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12

2 ln|| ln|| ln| 2|

ln|| ln l n 2

ln|| ln 2

Quick Review: Partial FractionDecomposition

1 2 2Let u=0;

1 2 1 2

1 2, 12Let u=-2

1 2, 12

BERNOULI EQUATIONS

P o Let

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can be reduced by letting where B0 find andsubstitute


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Differential Equations Notes Chapters 1-3

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