General Solutions of Linear Equations

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

If we assume 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 following 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.