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Higher Order Differential Equation&
Its Applications
Contents
IntroductionSecond Order Homogeneous DEDifferential Operators with constant coefficients
Case I: Two real roots Case II: A real double root Case III: Complex conjugate roots
Non Homogeneous Differential EquationsGeneral SolutionMethod of Undetermined CoefficientsReduction of OrderEuler-Cauchy EquationApplications
Introduction
A differential equation is an equation which contains the derivatives of a variable, such as the equation
Here x is the variable and a, b, c and d are constants.
Types of Differential EquationsHomogeneous DE
Non Homogeneous DE
0)()()()( 011
1
1
yxadxdyxa
dxydxa
dxydxa n
n
nn
n
n
)()()()()( 011
1
1 xgyxadxdyxa
dxydxa
dxydxa n
n
nn
n
n
Second Order Homogeneous DE
A linear second order homogeneous differential equation involves terms up to the second derivative of a function. For the case of constant multipliers, The equation is of the form
and can be solved by the substitution
Solution
The solution which fits a specific physical situation is obtained by substituting the solution into the equation and evaluating the various constants by forcing the solution to fit the physical boundary conditions of the problem at hand. Substituting gives
which leads to a variety of solutions, depending on the values of a and b. In physical problems, the boundary conditions determine the values of a and b, and the solution to the quadratic equation for λ reveals the nature of the solution.
Case I: Two real roots
For values of a and b such that
• there are two real roots m1 and m2 which lead to a general solution of the form
211 2( ) nm x m xm x
ny x c e c e c e
Case II: A real double root
If a and b are such that
then there is a double root λ =-a/2 and the unique form of the general solution is
Case III: Complex conjugate roots
For values of a and b such that
there are two complex conjugate roots of the form and the general solution is
The general solution of the non homogeneous differential equation
There are two parts of the solution:1. solution of the homogeneous part of DE
2. particular solution
( )ay by cy f x
cy
py
Non Homogeneous Differential Equations
General Solution of non-homogeneous equation is given by
represents solution of Homogeneous part represents particular solution
c py y y
General Solution
The method can be applied for the non – homogeneous differential equations , if the f(x) is of the form:
• constant
• polynomial function
•
•
• A finite sum, product of two or more functions of type (1- 4)
( )ay by cy f x
mxe
sin ,cos , sin , cos ,...x xx x e x e x
Method of Undetermined Coefficients
Reduction of Order
We know the general solution of is y = c1y1 + c2y1. Suppose y1(x) denotes a known solution of (1). We assume the other solution y2 has the form y2 = uy1.Our goal is to find a u(x) and this method is called reduction of order.
dxxy
exyydxxP
)()( 2
1
)(
12
Euler-Cauchy Equation
Form of Cauchy-Euler Equation
Method of SolutionWe try y = xm, since
)(011
11
1 xgyadxdyxa
dxydxa
dxydxa n
nn
nn
nn
n
k
kk
k dxydxa kmk
k xkmmmmxa )1()2)(1(
mk xkmmmma )1()2)(1(
Applications
Simple Harmonic Motion
Simple Pendulum
Applications (Cont.)
pressure change with altitude
Velocity Profile in fluid flow
Applications (Cont.)
vibration of springs
Electric circuits
Applications (Cont.)
Discharge of a capacitor
References
http://hyperphysics.phy-astr.gsu.edu/hbase/diff2.html#c2 http://tutorial.math.lamar.edu/Classes/DE/
IntroHigherOrder.aspx http://hyperphysics.phy-astr.gsu.edu/hbase/diff.html https://www.math.ksu.edu/~blanki/SecondOrderODE.pdf http://www.mylespaul.com/forums/showthread.php?t=266222
THANK YOU