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CHAPTER 3 : HIGHER — ORDER DIFFERENTIAL EQUATIONS
3.1 Preliminary Theory : Linear Equations
3.1.1 Initial Value and Boundary Value Problems
Initiat Value Problem:
Solve:
qo
i
0,62) 2 (3) Ee FIL 4a) = 8)
Subject t to:
yy) = Yor ¥ (Xp) = Vp VO? (XQ) = Vn
All initial conditions are specified at the same point Xo
Boundary Value Problem (BVP):
A BVP of second order
Solve:
a
a(x) a (a) 7 + ay(a)y = (2)
Subject to: :
YQ) = Yo, VCO) = Y,
Dependent Variable and its derivatives are specified at different points
May have several solutions, a unique solution or no solution
3.1.2 Homogeneous Equations
aml
a™y
4g, oo +a, (x) a
Feces +4,() 2 +0462) = ga)
If g(x)=0, homogeneous equation
If g(x)#0, then
qm
y
a
a, we asa, wt + 19,2) 2 +0962) =0
is called the associated homogeneous equation.
.Differdntial Operators
Dea | D™
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any
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a,Dy + --- +aDy 4£Q4Y2Oo
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Le aD 4----+ GD +a,
a lyse
Ea ya Sy" + by’ ~%y = Shy x
Ls D?s5D* 4D _y
= Ly) = sin x
«Superposition Principle for Homogeneous Equations
If y1,Yo....-Yk are k solutions of L{y)=0, then
CY) FCoYort....+Cy, is also a solution
Implications:
- y=0 is always a solution
- if y; is a solution of the homogeneous equation, then cy; is
also a solution
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Linear Dependence/Independence
A set of functions f;(x), fo(x),......fn(x) are linearly dependent on an interval | if
there exist constants Cj,...,C,, not all zero such that
ify (x) 4Cofo(x)+....... +Cpf,(x)=0 "xil
If the set is not linearly dependent, it is linearly independent.
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Wronskian: The determinant
f how f,
Wha frwo fi) = f f we f
feo od) a for
is called the Wronskian of the functions
Theorem: A set of functions is linearly independent if and only if
W(yi,¥2,---¥n)!0 for some range of x
Definition: A set of n linearly independent solutions of a linear ODE L(y)=0 is a
fundamental set of solutions.
Theorem: The general solution of the homogeneous ODE L(y)=0 is
Y=C1yi+CaVot..... bOnVn
where y;’s comprise a fundamental set and ¢;’s are constants
