Differential Equations Sections 4.1 Differential Equations

1
Differential Equations

• Sections 4.1

2
Differential Equations

• Recall
• A general solution is a family of solutions
• defined on some interval I that contains
• all solutions of the DE that are defined
• on I.
• In this chapter we will discuss finding
• general solutions of LDE of higher order
• than 1. We must investigate LDEs.

3
Differential Equations

• For LDE We will distinguish between
• Initial-Value and
• Boundary-Value Problems

4
Differential Equations

• A LDE n-th order Initial Value Problem
• Solve
• Subject to y(x0) y0, y(x0) y1 , , y(n-1)
  (x0) yn-1

5
Differential Equations

• Theorem 4.11
• Existence of a Unique Solution
• Let and g(x)
  be
• continuous on an interval I and let
• an(x) ? 0 for every x in this interval. If x0
• is any point in I, then the solution y(x) of
• the IVP exists on I and is unique.

6
Differential Equations

• Example 1 The given family of functions
• is the general solution of the DE on the
• indicated interval. Find a member of the
• family that is a solution of the IVP.

7
Differential Equations

• Example 2 Given that y c1 c2x2 is a
• two-parameter family of solutions of
• xy y 0 on the interval (-8, 8), show
• that constants c1 and c2 cannot be found
• so that a member of the family satisfies
• the initial conditions y(0) 0, y(0) 1.
• Explain why this does not violate Thm. 4.1.1

8
Differential Equations

• A LDE n-th order Boundary Value
• Problem
• Solve
• Subject to the dependent variable y or its
  derivatives are specified at different points.

9
Differential Equations

• So a 2nd order Boundary-Value Problem
• Solve
• Subject to y(a) y0, y(b) y1
• Or y(a) y0, y(b) y1
• Or y(a) y0, y(b) y1
• Or y(a) y0, y(b) y1

10
Differential Equations

• In the case of BVP, even if the conditions
• are satisfied as in Thm. 4.1.1, we may still
• have many, one or no solutions.

11
Differential Equations

• The given two-parameter family is a
• solution of the indicated DE on the
• interval (-8, 8). Determine whether a
• member of the family can be found that
• satisfies the boundary conditions.

12
Differential Equations

• Def A linear n-th order DE of the form
• is homogeneous, and
• is non-homogeneous with g(x) ? 0.

13
Differential Equations

• To solve a nonhomogeneous equation like
• in (2), we must solve the associated
• homogeneous equation (1).
• We will state for the remainder of the section
  that
• -The coefficient functions ai(x) for each i, and
  g(x) are continuous and
• - an(x) ?0 for every x in the interval.

14
Differential Equations

• Superposition Principle Homogeneous Eq.
• Let y1, y2, , yn be solutions of the
• homogeneous nth-order DE (1) on an
• interval I. Then the linear combination
• y c1 y1(x)c2 y2(x) cn yn(x)
• where ci are arbitrary constants, is also a
• solution on the interval.

15
Differential Equations

• Corollarys
• A constant multiple y c1y1(x) of a solution
  y1(x) of a homogeneous LDE is also a solution.
• A homogeneous LDE always possesses the trivial
  solution y 0.

16
Differential Equations

• Def Linear Dependence/Independence
• A set of functions f1(x), f2(x), , fn(x) is
• said to be linearly dependant on an
• interval I if there exist constants
• c1, c2, , cn, not all zero, such that
• c1f1(x) c2f2(x) cnfn(x)0
• For every x in the interval. If the set of
• fn. Is not linearly dependant then its
• linearly independent.

17
Differential Equations

• Linear dependence means for example for
• a set containing two functions that one is
• a constant multiple of the other.
• Note that f(x) x and g(x) sin(x) are
• linearly independent, since neither is
• simply a constant multiple of the other.

18
Differential Equations

• Theorem 4.1.3 Criterion for Linearly
• Independent Solutions.
• Let y1, y2, , yn be n solutions of the
• homogeneous liner nth-order DE (1) on an
• interval I. Then the set of solutions is
• linearly independent on I iff
• W(y1, y2, , yn ) ? 0 for every x in the
• interval.

19
Differential Equations

• What is the W(f1, f2, , fn)?
• If f1, f2, , fn are fn such that each has
• at least n-1 derivatives. The determinant
• W(f1, f2, , fn)
• The Wronskian of the functions.

20
Differential Equations

• Recall the determinant computation

21
Differential Equations

• Example Determine whether the given
• set of functions is linearly independent on
• the interval (-8,8).
• f(x) x, g(x) x3 , h(x) 3x3 6x
• f(x) 7, g(x) sinx, h(x) cosx

22
Differential Equations

• Def Any set y1, y2, , yn of n linearly
• independent solutions of the homogeneous
• linear nth-order DE (1) on an interval I is
• said to be a fundamental set of solutions
• on the on the interval.
• Thm. 4.1.4 There exists a f.s. of solutions
• for the homogeneous linear nth-order DE
• (1) on an interval I.

23
Differential Equations

• Theorem 4.1.5 Let y1, y2, , yn be a
• fundamental set of solutions of (1) on an
• interval I. Then the general solution of (1)
• on the interval I is
• y c1f1(x) c2f2(x) cnfn(x),
• where ci are arbitrary constants.

24
Differential Equations

• Example Verify the given functions form
• a fundamental set of solutions of the
• differential equation on the indicated
• interval. Form the general solution.

25
Differential Equations

• Reference
• Differential Equations
• With Boundary-Value Problems
• Zill Cullen
• Seventh Edition