4. Linear de-1

Def. 12. Linear differential equation is called an equation of the form

( 19 )

where a coefficient and an absolute term are known functions.

Theorem 5. Integration of a linear differential equation (19) is reduced to sequential integration of two separable first order differential equations.

■Let’s find a solution of the equation (19) in the form of a product of two unknown functions ,

. ( 20 )

Differentiating and substituting the values of in the equation we have

.

We try to choose some function to annulate the expression . It leads us to two separable differential equations

, (*)

. (**)

Integration of the first equation (*) gives

We substitute the found value in the equation (**) and find its general solution,

.

Finally we get the general solution of the given equation (19), namely

■

Ex. 12. Integrate a differential equation .

The differential equation is that linear with . Following the theory we put

and so

.

We have integrate successively the next two differential equations

, (*)

. (**)

As to (*) we have ,

,

By arbitrariness of C we can write

.

Passing to the equation (**) we get

.

Now the general solution of the given differential equation is

.

Ex. 13. Integrate the next differential equation .

Let’s rewrite the equation in the follows way:

, , , .

We see that the differential equation is that linear with respect to the unknown function . Therefore we do as follows

, (*)

. (**)

Solving the first equation (*) we have

.

Putting , we integrate the equation (**), namely

.

The general solution of the given differential equation

.

Ex. 14 (funds flow [movement of funds]). Let is amount of funds at a moment t. Retirement of funds during a time interval equals , where is some retirement coefficient. Growth of funds during the time interval equals , where is some coefficient (0< <1 because of not all investments are putting in funds) and I is a known amount of investments during one year. Amount of funds at the moment of time equals

.

Rate of movement of funds at a moment t equals

.

We’ve got Cauchy problem for a differential equation

with an initial condition

.

Coefficients of the equation can be constant or known functions of t. The quantity I can be constant or a function of t. In these cases the differential equation is that linear. What is more, I can be a function of the nth power of K(t), and in this case the equation is that of Bernoulli (see lower).

5. Bernoulli de-1

Def. 13. Bernoulli differential equation is called the next first order equation

( 21 )

where n is an arbitrary real number distinct from 0 and 1 ( ).

For the equation is a separable one and for it is a linear one.

We can integrate Bernoulli equation as a linear one if we put

.

The second method of integration consists in reducing of the equation (21) to that linear. Let’s divide both its sides by ,

,

and then suppose

.

We’ll get

.

The latter equation is a linear one in z (x).

Ex. 15. Integrate a differential equation .

The equation is Bernoulli one, . Finding its solution as a product

we have

(*)

(**)