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

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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 y₁ = y₁(x), y₂ = y₂(x), y₃ = y₃(x), ...... are solutions to a linear homogeneous differential equation, then

y = c₁ y₁(x) + c₂ y₂(x) + c₃ y₃(x) + .......... ,

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

Theorem 3. A set of solutions y₁ = y₁(x), y₂ = y₂(x), ........... , y_n = y_n(x) of a linear homogeneous differential equation is said to be linearly independent if

c₁ y₁(x) + c₂ y₂(x) + ......... + c_n y_n(x) = 0 ,

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

c₁ = c₂ = c₃ = .............. = c_n = 0 .

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

Linear combination of functions. The function c₁ f₁(x) + c₂ f₂(x) + ... + c_n f_n(x) with arbitrary numerical values for the coefficients c₁, c₂, ... ,c_n is called a linear combination of the functions f₁(x), f₂(x), ... , f_n(x).

Linearly dependent and independent sets of functions. A set of functions f₁(x), f₂(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 x², 3x + 1, 3x²+ 6x + 2 and x³ is linearly dependent since

3x²+ 6x + 2 = 3(x²) + 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 f₁(x), f₂(x), ... ,f_n(x) to be linearly independent is that

c₁ f₁(x) + c₂ f₂(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 f₁(x), f₂(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) = a₁ f_j(x) + a₂ 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_if_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:

c₁ f₁(x) + c₂ f₂(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_ix_i = 0 is not possible if c 0. Thus the set of functions is linearly independent if and only if

c₁ f₁(x) + c₂ f₂(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 f₁(x), f₂(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 y₁ = y₁(x), y₂ = y₂(x), ........... , y_n = y_n(x) are n linearly independent solutions of a linear homogeneous differential equation, then

y = c₁ y₁(x) + c₂ y₂(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) a₀(x)y⁽ⁿ⁾ + a₁(x)y^{(n -1)} + ........ + a_{n -1}(x)y ^'+ a_n(x)y = G( x)

and y₁(x), y₂(x), .... , y_n(x) be n linearly independent solutions of the corresponding homogeneous equation

a₀(x)y⁽ⁿ⁾ + a₁(x)y^{(n -1)} + ........ + a_{n -1}(x)y ^'+ a_n(x)y = 0

Then the general solution of 3) is given by

y = c₁ y₁(x) + c₂ y₂(x) + ......... + c_n y_n(x) + y_p(x)

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

c₁ y₁(x) + c₂ y₂(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.