Differential Equations By Pui chor Wong September 18, 2004 DeVry Calgary for Math230

Introduction Differntial equation is an equation that contains a derivative or a differential. Example: y'=3x 2 +8x-6 Example: xdy=4xy+y 2 dx

Order The order of a differential equation is the highest derivative in the equation Example: y'+x 3 -x=4 is called a first order differential equation Example: xy'+y 2 y''=8 is called a second order differential equation

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

Solution A solution to a differential equation is a relationship between the variable and differentials that satisfes the equation In 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/x 2

Linear Differential Equation A general procedure to solve a first order linear differential equation A 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

Second and Higher Order Direct Integration by reduction of the order if possible Example y''=A(x) y'=  A(x)dx repeat until y appears on the left side of the equation Linear or nonlinear, homogeneous or non- homgeneous, too complicated focus on linear, constant coefficients higher order differential equation

Homogeneous Equations If Q(x)=0, it is called homogeneous If Q(x)!=0, it is called nonhomogeneous Rewrite equation using D operator y'' should be written as D 2 y y' should be written as Dy D opertor is not an algebraic quantity but can be treated as so.

General solution (D - p 1 )y 1 =0 means y 1 = C 1 e p1x (D - p 2 )y 2 =0 means y 2 = C 2 e p2x (D - p 1 )(D - p 2 )y=0 means y= C 1 e p1x +C 2 e p2x

Distinct, Repeated, Complex Distinct: refer to previous case Repeated: y = C 1 e -px + C 2 xe -px Complex: y = e -ax (C 1 cos(bx) + C 2 sin(bx))

Non-Homogeneous y = y c + y p y c is called complementary solution y p is called particular solution use previous method to find complementary solution by letting Q(x)=0 first The to find particular solution, choose if Q(x) is of the form x n, y p will be of the form A+Bx+Cx kx n Solving for undetermined coefficients A, B,.. C etc.

Particular solution.. If Q(x) is of the form ae bx, y p is of the form Ae bx If Q(x) is of the form axe bx, y p is of the form Ae bx +Bxe bx If Q(x) is of the form a cos(bx) and a sin(bx), then y p is of the form Asin(bx)+Bcos(bx) If Q(x) is of the form axcos(bx) or axsin(bx), then y p is of the form Asin(bx)+Bcos(bx)+Cxcos(bx)+Exsin(bx)