Point 3. Homogeneous equations Structure of the general solution of so lhde

Theorem 5. (structure of the general solution of the second order linear homogeneous differential equation (3)). If be two linearly independent solutions of LHDE (3), then its general solution has the next form

( 11 )

where are arbitrary constants.

■The function is a solution of the equation (3) for any values of because of linear property of solutions of LHDE. On the base of the definition 8 of the lecture No. 22 it’s necessary to prove that for any initial conditions (2) one can find values of to satisfy these conditions. But

,

and so it is a matter of compatibility of the next system of linear equations in

This system has unique solution because of its principal determinant

distinct from zero by virtue of linear independence of the solutions ■

Ex. 6. A function is the general solution of the equation (see Ex. 5).

So lhde with constant coefficients

Let be given the second order linear differential equation

( 12 )

with constant coefficients . We’ll seek its solutions of the form

( 13 )

where k is an unknown (real or complex) number. Finding the derivatives of the function (13)

and substituting the values of in the equation we get

( 14 )

Equation (14) is called the characteristic equation. We have to study three cases.

1. Roots of the characteristic equation are real distinct. We get two linearly independent solutions of the equation (12) that is

because of . Therefore the general solution of the equation (12) is

. ( 15 )

Ex. 7. .

The characteristic equation of the differential equation

has two distinct real roots , so the differential equation has two linearly independent solutions

and so its general solution is

.

2. Roots of the characteristic equation are real equal .

We have one solution , and we must find the second linear independent solution of the equation (12). By Vieta theorem

,

so the equation (12) has in this case the form

( 16 )

and possesses the evident solution

.

Indeed,

Substituting in the equation (16) we get

.

Solutions are linearly independent, for

.

Therefore the general solution of the equation (12) (of the equation (16) in this case) is

. ( 17 )

Ex. 8. .

The characteristic equation

has equal real roots , two linearly independent solutions

,

so its general solution is

.

3. Roots of the characteristic equation (14) are complex, .

We have two complex solution of the equation (12)

,

but we can find two real solutions. By Euler formula

we do the next transformations

.

By virtue of the property 3 of solutions of LHDE (see Lecture No.22, P. 1) the next two real functions

are linearly independent solutions of the equation (12) . The-refore its general solution is

.

Ex. 9. .

The characteristic equation

has complex roots

,

therefore the differential equation has two real linearly independent solutions

and the general solution

.

Ex. 10. For the known equation (Ex. 5, 6) the characteristic equation has complex roots . Therefore linearly independent solutions and the general solution of the differential equation are