chapter_5

Higher Order Linear Differential Equations I & II (Constant Coefficients)

Definition of Linear Differential Equations

A linear differential equation of order n over an interval I can be expressed in the following standard form:

[ a_n y^{(n)} + a_{n-1} y^{(n-1)} + ... + a_1 y' + a_0 y = Q(x) ]

In this equation, the coefficients ( a_0, a_1, ..., a_n ) along with the non-homogeneous term ( Q(x) ) are continuous real functions of the variable ( x ) on the interval I. The order of the equation is determined by the highest derivative present. For example, the equation ( y''' = ext{sin}(x) ) is classified as a linear differential equation of order 3.

Properties of Linear Differential Equations

Linear differential equations can also be represented in the equivalent format:

[ P_1(y^{(n)}) + P_2(y^{(n-1)}) + ... + P_n(y) + Q = 0 ]

In this context, the ( P_i ) terms represent continuous functions. It is crucial to distinguish linear equations from non-linear ones, which involve terms with variable coefficients or powers greater than one. An example that illustrates this distinction is the equation ( 2xy y' = ext{sin}(x) ), which, despite being of degree one, is non-linear due to the multiplication of variables involved.

Criteria for Linear Equations

To qualify as a linear equation, the following conditions must be satisfied:

  1. The variable ( y ) and its derivatives appear only in the first degree (i.e., no powers higher than one).

  2. The derivatives are not multiplied together (e.g., no terms like ( y' y'' )).

  3. The coefficients of the equation must be constants or functions of ( x ).

Differential Operators and Notation

A differential operator, denoted by ( D ), operates on a function ( y = f(x) ). It generates derivatives, where:

  • ( D^n ) signifies the nth derivative of the function ( y ) with respect to ( x ), and so forth.

  • A polynomial of differential operators can be formulated as:

[ f(D) = D^n + P_1 D^{(n-1)} + P_2 D^{(n-2)} + ... + P_n ]

When the non-homogeneous term ( Q
eq 0 ), the equation ( f(D)y = Q ) is classified as a linear and non-homogeneous differential equation. In contrast, if ( Q = 0 ) for all ( x ) in I, then the equation ( f(D)y = 0 ) is termed linear and homogeneous.

Existence and Uniqueness Theorem

According to this theorem, if the functions ( P_i ) and the term ( Q ) are continuous over an interval I, and at least at one point ( x_0 \in I ) there are specific constants (denoted as ( a_0 )), then the equation has a unique solution ( y = y(x) ) that is valid over the interval I.

Solutions to Linear Homogeneous Differential Equations

If ( y = y_1(x) ) is a known solution of the homogeneous equation, then any function of the form:

[ y = C_1 y_1(x) ]

is also a solution, where ( C_1 ) is a constant. More generally, if several functions ( y_1, ..., y_k ) are solutions of the homogeneous equation, the linear combination:

[ y = C_1 y_1 + C_2 y_2 + ... + C_k y_k ]

also yields a solution of the homogeneous equation ( f(D)y = 0 ). The solutions ( y_i ) are termed linearly independent if the equation:

[ C_1 y_1 + C_2 y_2 + ... + C_n y_n = 0 ]

has only trivial solutions (where all constants are zero).

Wronskian Condition

The Wronskian determinant, denoted as ( W(y_1, y_2, ..., y_n) ), is defined as follows:

[ W(y_1, y_2, ', \cdots ,', y_n) = \det \begin{pmatrix} y_1 & y_2 & \cdots & y_n \ y_1' & y_2' & \cdots & y_n' \ \vdots & \vdots & \ddots & \vdots \ y_1^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)} \end{pmatrix} ]

The functions are said to be linearly independent over the interval I if and only if the Wronskian( W(y_1, y_2, ..., y_n) ) is non-zero for all ( x \in I ).

General Solution of ( f(D)y=0 )

The general solution of a linear homogeneous differential equation can be represented as:

[ y = C_1 y_1 + C_2 y_2 + ... + C_n y_n ]

where the functions ( y_i ) are any linearly independent solutions. The nature of these solutions can change depending on the roots of the auxiliary equation ( f(m) = 0 ), which can yield various cases, such as:

  • Real distinct roots

  • Repeated roots

  • Complex conjugate pairs

Particular Integrals (P.I.) and Complementary Functions (C.F.)

For a non-homogeneous equation of the form ( f(D)y = Q ), the general solution is given by:

[ y = y_c + y_p ]

Here, ( y_c ) represents the complementary function related to the homogeneous part ( f(D)y = 0 ), while ( y_p ) signifies the particular integral, which can be determined through various specific methods or formulas relevant to the non-homogeneous term.

Summary of Important Definitions

  • Auxiliary Equation (A.E.): This is the algebraic equation derived from the linear differential equation, which is crucial for identifying the roots needed for constructing general solutions.

  • Complementary Function (C.F.): This denotes the solution to the homogeneous segment of the differential equation ( f(D)y = 0 ).

  • Particular Integral (P.I.): This refers to the specific solution addressing the non-homogeneous part of the equation ( f(D)y = Q ).

Note on Linear Differential Equations with Constant Coefficients

In this context, it is specified that the coefficients are constant real numbers and that ( Q(x) ) must be a continuous function over the interval. The general operator form plays a significant role in determining homogeneous solutions of the equations.