Properties of solution and construction of general solution

Linear differential equation of the second order with constant coefficients.

.

Solution.

;

;

;

;

;

When , and

.

Last formula represents the law of the costs of improving of the quality of living labor.

Definition 6.1. A differential equation which can be written in the form

,

where and are constants, is continuous on some interval function, is called linear differential equation of the second order with constant coefficients.

If the equation is called non-homogeneous, if - homogeneous.

Consider homogeneous equation

Definition 6.2. Two solutions and are called linear dependent solutions on the some interval, if their ratio is equal to constant, that is

.

In opposed case, when

,

and are called linear independent.

Example 6.1. Let functions and be defined on some interval. Check their liner dependence or independence.

1)

and are linear dependent.

2)

and are linear independent.

3) , , where and are constants;

a) : and are linear independent.

b) : and are linear dependent.

Theorem 6.1. Let function be a solution of equation, then, where is constant, also will be a solution of this equation.

Proof. Finding , and substituting into the equation we will have

The expression in the bracket is equal to zero, because is a root of equation, then

.

It means thatis solution of equation.

Theorem 6.2. Let functions and be the solutions of equation, then their sumwill be a solution of this equation.

Дата добавления: 2014-01-11