logo Your Math Help is on the Way!

More Math Help

Algebraic Symmetries
Radical Expressions and Equation
The Exponential Function
Math 1010-3 Exam #3 Review Guide
Rational Numbers Worksheet
Are You Ready for Math 65?
Solving Simultaneous Equations Using the TI-89
Number Theory: Fermat's Last Theorem
Course Syllabus for Intermediate Algebra
Solving Inequalities with Logarithms and Exponents
Introduction to Algebra Concepts and Skills
Other Miscellaneous Problems
Syllabus for Calculus
Elementary Linear Algebra
Adding and Subtracting Fractions without a Common Denominator
Pre-Algebra and Algebra Instruction and Assessments
Mathstar Research Lesson Plan
Least Common Multiple
Division of Polynomials
Counting Factors,Greatest Common Factor,and Least Common Multiple
Real Numbers, Exponents and Radicals
Math 115 Final Exam Review
Root Finding and Nonlinear Sets of Equations
Math 201-1 Final Review Sheet
Powers of Ten and Calculations
Solving Radical Equations
Factoring Polynomials
Section 8
Declining Price, Profits and Graphing
Arithmetic and Algebraic Structures
Locally Adjusted Robust Regression
Topics in Mathematics
Syllabus for Mathematics
The Quest To Learn The Universal Arithmetic
Solving Linear Equations in One Variable
Examples of direct proof and disproof
Algebra I
Quadratic Functions and Concavity
More on Equivalence Relations
Solve Quadratic Equations by the Quadratic Formula
Solving Equations and Inequaliti
MATH 120 Exam 3 Information
Rational Number Ideas and Symbols
Math Review Sheet for Exam 3
Linear Algebra Notes
Factoring Trinomials
Math 097 Test 2
Intermediate Algebra Syllabus
How to Graphically Interpret the Complex Roots of a Quadratic Equation
The General, Linear Equation
Written Dialog for Problem Solving
Radian,Arc Length,and Area of a Sector
Internet Intermediate Algebra
End Behavior for linear and Quadratic Functions
Division of Mathematics
161 Practice Exam 2
General linear equations
Algebraic Symmetries
Math 20A Final Review Outline
Description of Mathematics
Math 150 Lecture Notes for Chapter 2 Equations and Inequalities
Course Syllabus for Prealgebra
Basic Operations with Decimals: Division
Mathematics Content Expectations
Academic Systems Algebra Scope and Sequence
Syllabus for Introduction to Algebra
Syllabus for Elementary Algebra
Environmental Algebra
More Math Practice Problems
Intermediate Algebra
Syllabus for Linear Algebra and Differential Equations
Intermediate Algebra
Rational Expressions and Their Simplification
Course Syllabus for Intermediate Algebra
GRE Review - Algebra
Foundations of Analysis
Finding Real Zeros of Polynomial Functions
Model Academic Standards for Mathematics
Study Guide for Math 101 Chapter 3
Real Numbers
Math 9, Fall 2009, Calendar
Final Review Solutions
Exponential and Logarithmic Functions

Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

The General, Linear Equation

2.1.2 The Wronskian

The Wronskian determinant of a pair of functions is defined as follows:

Definition 2.1.3

We often write W (u, v; x) = W (x) for short. When we do this we need to
remember that we have a specific pair of functions u and v in mind, and that
the sign of the Wronskian changes if we interchange them.

Theorem 2.1.6 Let u and v be solutions of equation (2.7). If is any
point of [a, b] and the value of the Wronskian there, then, at any point x
of [a, b],

Proof: Differentiating W as defined in equation (2.16) gives W' = uv''−u''v.
Substituting for the second derivatives from the equation (2.7), satisfied by
each of u and v, then gives W' = −pW. The conclusion follows from this.

An immediate deduction from this is the following:

Corollary 2.1.1
The Wronskian of a pair of solutions of a linear, homogeneous
equation (2.3) either vanishes identically on the entire interval [a, b]
or does not vanish at any point of the interval.

The Wronskian is precisely the determinant appearing in the systems
(2.13) and (2.14) above. When the Wronskian vanishes, the solutions and
are linearly dependent, and when the Wronskian does not vanish they
are linearly independent and therefore constitute a basis. This proves the

Theorem 2.1.7 A necessary and sufficient condition for a pair of
solutions of equation (2.7) to be a basis for that equation is that its Wronskian
W not vanish on [a, b].

Remark: If a pair of solutions and is chosen “at random,” i.e., if
their initial data are assigned randomly, we would expect that W ≠ 0; in
this sense it would take a “peculiar” choice of solutions for that pair to be
linearly dependent.

The expression (2.17) for the Wronskian may be used to find a formula
for a second solution v of equation (2.7) if we know one solution u. Suppose
one such solution u (not identically zero) is known on [a, b]. Pick a point
where u ≠ 0 and define

where c is not zero but is otherwise arbitrary. Write

so that

This formula defines v on an interval containing on which u does not vanish.
It is straightforward to check that v is indeed a solution of equation (2.7)
on such an interval, and that the pair u, v has the nonvanishing Wronskian
w there.


1. Verify equation (2.2) in detail for the special case when n = 2 and the
operator L is defined by equation (2.5). Using equation (2.2), verify Lemma

2. Let L be the second-order operator defined by equation (2.5). Suppose k
functions (x) , i = 1, . . . , k are given on the interval [a, b] and the corresponding
k initial-value problems

are solved for the k functions , i = 1, . . . , k. Define .
Show that the unique solution of the initial value problem Lu = r, u () =
u' () = 0 is

3. In equation (2.7) let the interval be [-1, 1]. Can continuous coefficients q
and p be chosen such that u (x) = x2 is a solution of this equation?

4. Let u (x) = x3 on [-1, 1] and define v (x) = -x3 on [-1, 0] and v (x) = +x3
on (0, 1]. Verify that v is C2 on [-1, 1]. Calculate W (u, v; x). Are these
functions linearly dependent on [-1, 1]?

5. The fundamental theorem of algebra states that a polynomial of degree n
can have no more than n distinct zeros. Draw the conclusion of Example
2.1.1 from this.

6. Show that the three functions sin(x), sin(2x), sin(3x) are linearly independent
on any nontrivial interval of the x axis.

7. For the equation u''+u = 0, cos x and sin x form a basis of solutions. Verify
that cos (x + a) and sin (x + a) are also solutions if a is any constant. Use
this and the uniqueness theorem to infer the trigonometric addition theorems

cos (x + a) = cos a cos x - sin a sin x,
sin (x + a) = cos a sin x + sin a cos x.

8. Find bases of solutions for the following equations:

a) u'' = 0;
b) u'' + 2u' = 0;
c) u'' + xu' = 0.

9. Consider the equation u'' + u = 0. Under what conditions on the constants
a, b, c, d is the pair

a basis of solutions for this equation?

10. Consider the equation u'' - u = 0. Under what conditions on the constants
a, b, c, d is the pair

a basis of solutions for this equation?

11. Show that the functions and form a basis of
solutions for the equation

on any interval excluding the origin.

12. Same as the preceding problem for the functions = sin x − x cos x , =
cos x + x sin x and the differential equation

13. Verify that the second solution given in the formula (2.18) is linearly independent
of u on intervals on which u does not vanish.

14. Let u be a solution of (2.7) vanishing at the endpoints and of an
interval but not vanishing in between. Show that the formula (2.18), with
, leads to finite limiting values for the second solution v at
and , and find these limits.

15. A second-order equation

is said to be in ”self-adjoint” form. What condition on the function p is
needed for the existence theorem 2.1.1 to hold? Show that the Wronskian
of such a self-adjoint equation is, except for a multiplicative constant, p-1.

16. For the equation

• On what interval does the existence theorem guarantee a solution?

• Verify that = x is a solution.

• Find a second solution in an interval containing the origin. How far
can this interval extend on each side of the origin?

17. Let u and v be C2 functions on an interval I, and suppose their Wronskian
uv' − u'v does not vanish on I. Show that their zeros separate each other,
i.e., between any two consecutive zeros of u there is exactly one zero of v
and vice versa.

18. Consider the equation u'' +p(x)u' +q(x)u = 0 where p and q are continuous
on the entire real axis R. Suppose further that q(x) < 0 there. Show that if
u is not identically zero, it can have at most one zero on R.

19. Let u and v be given C2 functions on an interval [a, b] whose Wronskian
nowhere vanishes there. Show that there is a differential equation of the
form (2.7) for which u and v form a basis of solutions.

20. Consider the initial-value problem on the interval [−1, 1]

u'' + q (x) u = 0, u (−1) = 1, u' (−1) = 0,

where the coefficient q is the sectionally continuous function

Find a function u satisfying the initial conditions, C1 on the interval [−1, 1],
and satisfying the equation on the intervals [−1, 0) and (0, 1].

21. Suppose p and q are continuous functions on the symmetric interval [−a, a]
and satisfy the further conditions

p(−x) = −p(x) and q(−x) = q(x)

there; i.e., p is an odd function and q is even. For the differential equation
u'' + p(x)u' + q(x)u = 0 show

(a) If u(x) is a solution then so is v(x) = u(−x).

(b) There exists a basis of solutions , of which one is even on the
interval, the other odd.