Math 260

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Math 3120 Differential Equations withBoundary
Value Problems
Chapter 4 Higher-Order Differential
Equations Section 4-1 Preliminary Theory-Linear
Equations
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Initial-Value and Boundary-Value Problems

• Theorem 4.1.1 An initial value problem for a
  nth order Linear DE has the general form
• subject to the initial conditions
• If the functions P0,, Pn, and G are continuous
  on an open interval I (?, ? ), and that P0 is
  nowhere zero on I. If t t0 is any point in I,
  then there exists exactly one solution y ?(t)
  that satisfies the initial value problem. This
  solution exists throughout the interval I .

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Example 1

• Verify that is a
  unique solution to the IVP

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Boundary-Value Problems

• Is a linear DE of order two or greater in which
  the dependent variable y or its derivative is
  specified at different points. For example,
• The values are
  called boundary conditions.
• A solution to Eq. (1) is a function y ?(x) that
  satisfies the differential equation on the
  interval ? lt x lt ? and that takes on the
  specified values y0 and y1 at the endpoints.

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Example 2 BVP has a unique solution

• Consider the boundary value problem
• The general solution of the differential equation
  is
• The first boundary condition requires that c1
  1.
• From the second boundary condition, we have
• Thus the solution to the boundary value problem
  is
• This is an example of a nonhomogeneous boundary
  value problem with a unique solution.

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Example 3 BVP has No solution

• Consider the boundary value problem
• The general solution of the differential equation
  is
• The first boundary condition requires that c1
  1, while the second requires c1 - a. Thus
  there is no solution.

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Example 4 BVP has Infinite solution

• Consider the boundary value problem
• The general solution of the differential equation
  is
• However, if a -1, then there are infinitely
  many solutions
• This example illustrates that a non homogeneous
  boundary value problem may have no solution, and
  also that under special circumstances it may have
  infinitely many solutions.

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Homogeneous Equations

• An nth order homogenous Linear DE is of the form
• An nth order nonhomogenous Linear DE is of the
  form
• When we talk about Linear equations of form (1),
  we will assume that the coefficient functions
  Pi(t), i 0,1,n and G(t) are continuous and
  P0(t) ? 0 for every t in the interval.

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Differential Operator

• We can denote differentiation by the Capital
  letter D, i.e.,
• This symbol D is called the Differential
  operator. Higher-Order derivatives can also be
  expressed in terms of D. For example,
• An nth order differential operator or polynomial
  operator is defined as
• L is a linear operator, i.e., Laf(x)ßg(x)
  aL(f(x))ßL(g(x))

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Theorem 4.1.2 Superposition Principle

• If y1,, yn are solutions of the homogeneous nth
  order linear differential equation on an interval
  I
• Then the linear combination
• where the ci , i 0,1,,n are arbitrary
  constants is also a solution on the interval I.

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Linear Independence and Linear Dependence

• A set of functions f1(x), f2(x),., fn(x) is said
  to be linearly dependent on an interval I if
  there exist constants c1, c2,,cn, not all zero,
  such that
• for every x in the interval.
• If the set of functions is not linearly dependent
  on the interval I, then it is said to be linearly
  independent.

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Wronskian

• Suppose each of the functions f1(x), f2(x),.,
  fn(x) possesses at least n-1 derivatives. The
  determinant
• where the primes denote derivatives, is called
  the Wronskian of the functions.

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Homogeneous Equations Wronskian

• If y1,, yn are solutions of the homogeneous nth
  order linear differential equation on an interval
  I , then the set of solutions is Linearly
  independent iff Wronskian, is nonzero at t0 in
  the interval I.
• Since t0 can be any point in the interval I, the
  Wronskian determinant needs to be nonzero at
  every point in I.
• As before, it turns out that the Wronskian is
  either zero for every point in I, or it is never
  zero on I.

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Fundamental Solutions Linear Independence

• Consider the nth order homogenous LDE
• A set y1,, yn of solutions of Eq. (3) with
  W(y1,, yn) ? 0 on I is called a fundamental set
  of solutions.
• If y1,, yn are fundamental solutions, then
  W(y1,, yn) ? 0 on I. It can be shown that this
  is equivalent to saying that y1,, yn are
  linearly independent

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General Solution of Homogenous Equations

• Since all solutions can be expressed as a linear
  combination of the fundamental set of solutions,
  the general solution of the equation on the
  interval is
• where cis are arbitrary constants

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Example 5

• Verify that the given functions are solutions of
  the differential equation, and determine their
  Wronskian.

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Nonhomogeneous Equations

• Consider the non homogeneous equation
• Any function yp that has no arbitrary parameters
  and satisfies the Eq. (4) is said to be a
  particular solution of the non homogeneous
  equation.
• The general solution to the non homogeneous
  equation (4) is
• where yp is any particular solution and y1,, yn
  are the fundamental solutions of the associated
  homogeneous differential equation (3) on the
  interval I and where cis are arbitrary constants.

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Theorem 4.1.7 Superposition Principle Non
Homogenous Equations

• If yp1,, ypk be k particular solutions of the
  non homogeneous nth order linear differential
  equation (4) on an interval I corresponding, in
  turn, to k distinct functions g1,, gk, i.e.,
• where i1,.,k. Then
• is a particular solution of (5).