8

S.B. Karavashkin

The presented solutions emphasise again a vast diversity of vibration forms and patterns dependently on the line type and parameters.

Check the correspondence of presented solutions (13), (16) and (17) to (12). It will be sufficient to substitute them sequentially into the first, arbitrary i th and the last equations of this system.

For the periodic regime on the left part of the first equation of (12) we yield

and on the right part

For the left part of the i th equation

and for the right part

For the left part of the last equation of the system

and for the right part

Thus, the periodic solution satisfies (12) completely.

For the aperiodical regime it is sufficient to check, whether the solutions satisfy any equation of the system, because, as we showed in [1], this regime, the same as that critical, is one of the forms of the common solution, and if the periodical solution satisfied the system, then the aperiodical and critical ones satisfy too. So make a control check - for example, for the first, the most complicated equation of the system.

For its left part we yield

and for the right one

Similarly, on the left part of the first equation for critical regime

and on the right part

As we can see, if the solutions were complete and exact, they satisfy as the general equation of the system as the starting and finishing equations as the equation containing the external force. With it, we can see from the equations reduction pattern that the way to reduce is different in each case, because in the solutions themselves their own specific multipliers corresponding to each equation of the modelling system are present; in this way they provide the identity in each case. This is one more interesting property of the obtained solutions, whence it follows that one and the same solution satisfies few types of differential equations at the same time. This was the main complication and main reason of failures when many generations of scientists sought the exact solutions.