SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

General solution of n-th order linear differential equations. Linearly dependent and independent sets of functions, Wronskian test for dependence.

General n-th order linear differential equation. The general n-th order linear differential equation is an equation of the form

where the a_{i}(x) are functions of x only. If G(x) = 0, the resulting equation

is called a homogeneous linear differential equation, the term homogeneous being used to indicate that it is homogeneous with respect to y and its derivatives y', y'', ... (the functions in x not being considered). Equation 1) is called a nonhomogeneous equation.

Theorems on solutions to linear homogeneous differential equations

Theorem 1. If y = f(x) is a solution to a linear homogeneous differential equation, then y = cf(x), where c is an arbitrary constant, is also a solution.

Theorem 2. If y_{1} = y_{1}(x), y_{2} = y_{2}(x), y_{3} = y_{3}(x), ...... are solutions to a linear homogeneous
differential equation, then

y = c_{1} y_{1}(x) + c_{2} y_{2}(x) + c_{3} y_{3}(x) + .......... ,

where the c’s are arbitrary constants, is also a solution.

Theorem 3. A set of solutions y_{1} = y_{1}(x), y_{2} = y_{2}(x), ........... , y_{n} = y_{n}(x) of a linear
homogeneous differential equation is said to be linearly independent if

c_{1} y_{1}(x) + c_{2} y_{2}(x) + ......... + c_{n} y_{n}(x) = 0 ,

(where the c’s are arbitrary constants) holds only under the condition that

c_{1} = c_{2} = c_{3} = .............. = c_{n} = 0 .

Linearly dependent and independent sets of functions, Wronskian test for dependence

Linear combination of functions. The function c_{1} f_{1}(x) + c_{2} f_{2}(x) + ... + c_{n} f_{n}(x) _{ } with
arbitrary numerical values for the coefficients c_{1}, c_{2}, ... ,c_{n} is called a linear combination of the
functions f_{1}(x), f_{2}(x), ... , f_{n}(x).

Linearly dependent and independent sets of functions. A set of functions f_{1}(x),
f_{2}(x), ... ,f_{n}(x) is said to be linearly dependent if some one of the functions in the set can be
expressed as a linear combination of one or more of the other functions in the set. If none of the
functions in the set can be expressed as a linear combination of any other functions of the set,
then the set is said to be linearly independent.

Example. The set of four functions x^{2}, 3x + 1, 3x^{2}+ 6x + 2 and x^{3} is linearly dependent since

3x^{2}+ 6x + 2 = 3(x^{2}) + 2(3x + 1)

A necessary and sufficient condition for the linear independence of a set of functions. There exists an important algebraic criterion, an algebraic test, which can tell us whether a set of functions is linearly independent or not. That test is given by the following theorem:

Theorem. A necessary and sufficient condition for the set of functions f_{1}(x), f_{2}(x), ...
,f_{n}(x) _{ }to be linearly independent is that

c_{1} f_{1}(x) + c_{2} f_{2}(x) + ... + c_{n} f_{n}(x) = 0

only when all the scalars c_{i} are zero.

What is the reasoning that leads to the assertion of this theorem? Well, a set of functions f_{1}(x),
f_{2}(x), ... ,f_{n}(x) is linearly dependent if some one of the functions in the set can be expressed as a
linear combination of one or more of the other functions in the set, that is if there exists some
function f_{i}(x) in the set such that

f_{i}(x) = a_{1} f_{j}(x) + a_{2} f_{k}(x) + ...

for one or more functions f_{j}(x), f_{k}(x) , etc. of the set. This condition implies that there exists
some subset of functions f_{i}(x), f_{j}(x), f_{k}(x), etc. within the full set such that

c_{i }f_{i}(x) + c_{j} f_{j}(x) + c_{k} f_{k}(x) + ... = 0

where c_{i} c_{j}, c_{k}, etc. are non-zero. Said differently, a set is linearly dependent if there exist two or
more non-zero c’s for which the following equation holds true:

c_{1} f_{1}(x) + c_{2} f_{2}(x) + ... + c_{n} f_{n}(x) = 0

(i.e. it is possible for the equation to hold true even though not all of the c’s are zero). If there
does not exist two or more non-zero c’s for which it will hold, then the set of functions is linearly
independent. The case in which only one of the c’s is non-zero is impossible since c_{i}x_{i} = 0 is not
possible if c
0. Thus the set of functions is linearly independent if and only if

c_{1} f_{1}(x) + c_{2} f_{2}(x) + ... + c_{n} f_{n}(x) = 0

only when all the scalars c_{i} are zero.

Wronskian test for dependence. A test for the linear dependence of a set of n functions
f_{1}(x), f_{2}(x), ... , f_{n}(x) having derivatives through the (n-1)th order can be obtained through
evaluation of the Wronskian determinant

If the Wronskian is not identically zero, the functions are linearly independent. If it is identically zero over an interval (a, b), the functions are linearly dependent on the interval.

Theorem 4. A necessary and sufficient condition that a set of solutions of a linear homogeneous differential equation be linearly independent is that the Wronskian not be identically zero.

General solution of a linear homogeneous differential equation. If y_{1} = y_{1}(x), y_{2} =
y_{2}(x), ........... , y_{n} = y_{n}(x) are n linearly independent solutions of a linear homogeneous
differential equation, then

y = c_{1} y_{1}(x) + c_{2} y_{2}(x) + ......... + c_{n} y_{n}(x)

is the general solution of the equation.

General solution of an n-th order linear differential equation. Let y_{p}(x) be any
particular solution of the n-th order linear differential equation

3) a_{0}(x)y^{(n)} + a_{1}(x)y^{(n -1)} + ........ + a_{n -1}(x)y ^{' }+ a_{n}(x)y = G( x)

and y_{1}(x), y_{2}(x), .... , y_{n}(x) be n linearly independent solutions of the corresponding homogeneous
equation

a_{0}(x)y^{(n)} + a_{1}(x)y^{(n -1)} + ........ + a_{n -1}(x)y ^{' }+ a_{n}(x)y = 0

Then the general solution of 3) is given by

y = c_{1} y_{1}(x) + c_{2} y_{2}(x) + ......... + c_{n} y_{n}(x) + y_{p}(x)

The portion of the general solution that comes from the associated homogeneous equation i.e.

c_{1} y_{1}(x) + c_{2} y_{2}(x) + ......... + c_{n} y_{n}(x)

is called the complementary function. The particular solution y_{p}(x) is also called a particular
integral. Thus the general solution of the general equation consists of the sum of the
complementary function and a particular integral.

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]