Differential Equations

By Pui chor WongSeptember 18, 2004DeVry Calgaryfor Math230

Introduction

Differntial equation is an equation that contains a derivative or a differential.Example: y'=3x2+8x-6Example: xdy=4xy+y2dx

Order

The order of a differential equation is the highest derivative in the equationExample: y'+x3-x=4 is called a first order differential equationExample: xy'+y2y''=8 is called a second order differential equation

Degree

The degree of a differential equation is the power of the highest order derivativeExample xy''+y2y'-3y=6 is called the first degree second order differential equationExample (y''')2+3y'=0 is a second-degree, third-order differential equation

SolutionA solution to a differential equation is a relationship between the variable and differentials that satisfes the equationIn general, a differential equation has an infinite family of solutions. That is called general solution.The solution of an nth-order differential equation can have at most n arbitrary constants.A solution having the maximum number of constants is called the general solution or complete solution.When additional information is given to determine at least one of the conditions, the solution is then called particular solution

Separation of Variables

One kind of DE (differential equation) can be solved by separating the variables and integrate.A first-order, first degree differential equation y'=f(x,y) is called separable if it can be written in the form y'=A(x)/B(x) or A(x)dx=B(y)y

Integrable Combinations

Consider the product rule or quotient rule d(xy) = xdy+ydx d(x/y) = xdy-ydx/x2

Linear Differential Equation

A general procedure to solve a first order linear differential equationA first-order differential equation is aid to be linear if it can be written in the form y' + P(x)y = Q(x)

Solution using integrating factor

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Second and Higher Order

Direct Integration by reduction of the order if possibleExample y''=A(x) y'=A(x)dx repeat until y appears on the left side of the

equationLinear or nonlinear, homogeneous or non-homgeneous, too complicatedfocus on linear, constant coefficients higher order differential equation

Homogeneous Equations

If Q(x)=0, it is called homogeneousIf Q(x)!=0, it is called nonhomogeneousRewrite equation using D operatory'' should be written as D2yy' should be written as DyD opertor is not an algebraic quantity but can be treated as so.

General solution

(D - p1)y1=0 means y1= C1 e p1x

(D - p2)y2=0 means y2= C2 e p2x

(D - p1)(D - p2)y=0 means y= C1 e p1x+C2 e p2x

Distinct, Repeated, Complex

Distinct: refer to previous caseRepeated: y = C1e-px + C2xe-px

Complex: y = e-ax(C1cos(bx) + C2sin(bx))

Non-Homogeneous

y = yc + yp

yc is called complementary solution

yp is called particular solution

use previous method to find complementary solution by letting Q(x)=0 firstThe to find particular solution, choose if Q(x) is of the form xn, yp will be of the form A+Bx+Cx2+..+kxn

Solving for undetermined coefficients A, B,.. C etc.

Particular solution..If Q(x) is of the form aebx, yp is of the form Aebx

If Q(x) is of the form axebx, yp is of the form Aebx+Bxebx

If Q(x) is of the form a cos(bx) and a sin(bx), then yp is of the form Asin(bx)+Bcos(bx)

If Q(x) is of the form axcos(bx) or axsin(bx), then yp is of the form Asin(bx)+Bcos(bx)+Cxcos(bx)+Exsin(bx)