Liner first order differential equations

Definition 4.1. Differential equation is called linear of the first order if it is equation of the first order with respect to and .

, were -are continuous functions and .

Let us divide this equation by

.

Denote and we get equation

Definition 4.1. A differential equation that can be written in the form

is called a linear first order differential equation.

Let us represent unknown function in the form

and find its derivative

On substituting and into the equation we get

or

.

Now we take as particular solution of equation in the brackets

On separating variables we obtain

whence

,

or

To determine the function we have equation

.

On solving this equation

we find the unknown function

Substituting and in we will write the general solution of the linear equation

Example 4.1. Find general solution of the differential equation

.

Solution. Let us write this equation in form

.

We will find the solution as a product of two unknown functions and .

;

;

;

.

After separating variables

;

;

;

;

;

;

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