The general linear ordinary differential equation of order
n has the form

If we as sume that on
the interval I we can reduce this equation to

the form

If the function f ≡ 0 we say that the equation (2)
is homogeneous:

**Theorem 1. **Let
be n solutions of the homogeneous linear equation

(3) on the interval I. If are constants then
the linear combination

is also solution to equation (3).

.

The fol lowing theorem provide the existence and uniqueness result for the

initial value problem associated with the equation (2).

**Theorem 2.** Suppose that the functions
and f are continuous

on the open interval I containing the ppoint a. Then, given n numbers

the nth-order linear equation (2)

has a unique solution on the interval I that satisfies the
n initial conditions

**Definition. **The n functions
are said to be linearly dependent

on the interval I provided that there exist constants
not

all zero such that

on I.

Suppose that are
solutions to the homogeneous equation (3).

If they are linearly independent then the general solution to equation (3) is

given by formula :

Denote by some
solution to equation (2). Let
are linearly
independent

solutions to homogeneous equation (3) . Then the general solution

to equation (3) is

**Example 1.** Find a particular solution satisfying
the given initial conditions

Solution. the general solution to the ordinary
differential equation is

Let’s find the first and the second order derivatives of
this solution

and

We have

One can rewrite this system in the form

We subtract from the second and third equations the first
one

Therefore

Finally we have

The solution to the initial value problem is

**Example 2. **Find a solution satisfying the given
initial condition

Solution. The general solution to the homogeneous ordinary
differential

equation y'' + y = 0 is

Hence the general solution to our equation is

Taking the derivative of the function y'(t) we obtain

Then

from the first equation we have

and from the second one

Hence the solution is

**Example 3.** Use the Wronskian to prove that
functions f(x) = e^{x}, g(x) =

e^{2x}, h(x) = e^{3x} are linearly independent on the real line .

Solution.