Find a General Solution to the Following Higher-order Equations
Higher Order Equations with Constant Coefficients
This section is devoted to an introductory study of higher order linear equations with constant coefficients. This is an extension of the study of order linear equations with constant coefficients (see, Section 8.3).
The standard form of a linear order differential equation with constant coefficients is given by
(8.6.1) |
where
is a linear differential operator of order with constant coefficients, being real constants (called the coefficients of the linear equation) and the function is a piecewise continuous function defined on the interval We will be using the notation for the derivative of If then (8.6.1) which reduces to
(8.6.2) |
is called a homogeneous linear equation, otherwise (8.6.1) is called a non-homogeneous linear equation. The function is also known as the non-homogeneous term or a forcing term.
DEFINITION 8.6.1 A function defined on is called a solution of (8.6.1) if is times differentiable and along with its derivatives satisfy (8.6.1).
As in Section 8.3, we first take up the study of (8.6.2). It is easy to note (as in Section 8.3) that for a constant
where,
(8.6.3) |
DEFINITION 8.6.3 (Characteristic Equation) The equation where is defined in (8.6.3), is called the CHARACTERISTIC EQUATION of (8.6.2).
Note that is of polynomial of degree with real coefficients. Thus, it has zeros (counting with multiplicities). Also, in case of complex roots, they will occur in conjugate pairs. In view of this, we have the following theorem. The proof of the theorem is omitted.
THEOREM 8.6.4 is a solution of (8.6.2) on any interval if and only if is a root of (8.6.3)
- If are distinct roots of then
- If is a repeated root of of multiplicity i.e., is a zero of (8.6.3) repeated times, then
- If is a complex root of then so is the complex conjugate Then the corresponding linearly independent solutions of (8.6.2) are
EXAMPLE 8.6.5
- Find the solution space of the differential equation
Solution: Its characteristic equation is - Find the solution space of the differential equation
Solution: Its characteristic equation is - Find the solution space of the differential equation
Solution: Its characteristic equation is
From the above discussion, it is clear that the linear homogeneous equation (8.6.2), admits linearly independent solutions since the algebraic equation has exactly roots (counting with multiplicity).
DEFINITION 8.6.6 (General Solution) Let be any set of linearly independent solution of (8.6.2). Then
is called a general solution of (8.6.2), where are arbitrary real constants.
EXAMPLE 8.6.7
- Find the general solution of
Solution: Note that 0 is the repeated root of the characteristic equation So, the general solution is - Find the general solution of
Solution: Note that the roots of the characteristic equation are So, the general solution is
EXERCISE 8.6.8
- Find the general solution of the following differential equations:
- Find a linear differential equation with constant coefficients and of order which admits the following solutions:
- and
- and
- and
- Solve the following IVPs:
- Euler Cauchy Equations:
Let be given constants. The equation(8.6.4)
is called the homogeneous Euler-Cauchy Equation (or just Euler's Equation) of degree (8.6.4) is also called the standard form of the Euler equation. We define(8.6.5)
Essentially, for finding the solutions of (8.6.4), we need to find the roots of (8.6.5), which is a polynomial in With the above understanding, solve the following homogeneous Euler equations: - Consider the Euler equation (8.6.4) with and Let or equivalently Let and Then
- show that or equivalently
- using mathematical induction, show that
- with the new (independent) variable , the Euler equation (8.6.4) reduces to an equation with constant coefficients. So, the questions in the above part can be solved by the method just explained.
We turn our attention toward the non-homogeneous equation (8.6.1). If is any solution of (8.6.1) and if is the general solution of the corresponding homogeneous equation (8.6.2), then
is a solution of (8.6.1). The solution involves arbitrary constants. Such a solution is called the GENERAL SOLUTION of (8.6.1).
Solving an equation of the form (8.6.1) usually means to find a general solution of (8.6.1). The solution is called a PARTICULAR SOLUTION which may not involve any arbitrary constants. Solving (8.6.1) essentially involves two steps (as we had seen in detail in Section 8.3).
Step 1: a) Calculation of the homogeneous solution and
b) Calculation of the particular solution
In the ensuing discussion, we describe the method of undetermined coefficients to determine Note that a particular solution is not unique. In fact, if is a solution of (8.6.1) and is any solution of (8.6.2), then is also a solution of (8.6.1). The undetermined coefficients method is applicable for equations (8.6.1).
A K Lal 2007-09-12Find a General Solution to the Following Higher-order Equations
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