University Mathematics Handbook (2015)

XI. Ordinary Differential Equations, or ODE

Chapter 2. Linear n-th Order Differential Equations

2.1  Definition

a.  The equation

(*)

is a -th order linear equation.

b.  If the equation is called a homogeneous linear equation.

c.  If all coefficients are continuous functions on , then in the neighborhood of initial conditions

, (**)

the Cauchy's problem (*, **) has a unique solution.

2.2  Linear Operator

a.  We denote

b.  For every fixed :

c.  

d.  If and are solutions of the equation , then is also a solution of that equation.

e.  If ODE with real coefficients has complex solution , then and are real solutions of that equation.

f.  If is a non-trivial solution of then, by substituting and , we get an order ODE for .

2.3  Solutions of

a.  Functions are linearly dependent (LD) in interval if there exist numbers , not all of which are zeroes, such that . Otherwise, functions are linearly independent (LI).

b.  If functions are LD on , then the determinant

is identically equal to zero.

Determinant is called Wronskian.

c.  If LI functions are solutions of with continuous coefficients on , then Wronskian is zero nowhere on .

d.  Particular solutions of , with a continuous coefficient, which are linear independent on , form a basis of the solution space.

e.  Any solution of is a linear combination of LI solutions

2.4  Restoring Linear ODE

a.  Let be LI functions in .

We construct an th order determinant

Developing it by its last column, we get a linear ODE the solutions of which are the given functions.

b.  Abel's identity formula:

2.5  Linear Homogeneous ODE with Constant Coefficients

Let

be the n-th-order linear ODE, when are its constant real coefficients. Then, the solutions are in the form of . Substituting, we get the characteristic equation

a.  If the characteristic equation has different real solutions , then are LI solutions forming a basis of solution space. The general solution is when are constants.

b.  If the characteristic equation has complex solution , then is also one of its solutions. It is written . Therefore, this ODE has two LI solutions, and .

c.  If a characteristic equation has a root with multiplicity , then it fits LI solutions .

d.  If a characteristic equation has complex root with multiplicity , then it has real LI solutions:

2.6  Euler's Equation

The equation

when are real constant coefficients, is called Euler's equation.

a. The substitution transforms Euler's equation into a constant coefficient linear ODE.

b. We look for a solution in the form of . The result is a characteristic equation with respect to :

c. For each real root with multiplicity we get LI solutions

d. For each pair of complex roots with multiplicity we get LI solutions:

2.7  Non-homogeneous Linear ODE

a.  The general solution of the ODE

(*)

is when is the general solution of the homogeneous equation and is a particular solution of the non-homogeneous equation, respectively.

b.  The parameter variation method of finding a particular solution:

If is a general solution of (*) in the homogeneous equation, we look for a solution of the following form:

Substituting it in equation (*), we construct a linear equation system with respect to :

The determinant of the system is Wronskian . Therefore, the system has a unique solution.

2.8  Non-homogeneous Linear ODE with Constant Coefficients

(*)

Characteristic equation fitting (*)

(**)

If is one of the following functions:

1. Polynomial

2.

3. or

Then, we can find a particular solution of (*) on the form presented in the table, where is a given polynomial, and are unknown polynomials, and are real numbers: