The matter belongs to the solution of the uniform linear differential equations of n-go of an order. At the same time is proved, but search of the system of decisions called fundamental (for short FSR) hesitates on concrete examples, which linear combination of functions gives the common decision of the differential equation.
Instruction
1. The differential equation of the highest order is called linear if it linearly rather unknown function and all its derivatives. The general view of the linear uniform differential equation (LUDE) of n-go of an order illustrates fig. 1.
2. The left member of equation (1) is called linear differential operator n-go of an order and designated: L [y]: L [y] =y^ (n) +a1 (x) y^(n-1)+ … +a(n-1) (x) y '+a^n (x y) =0. The equation (1) it is possible to rewrite in the form of L [y]=0.
3. Let on an interval (a, b) is given the system of functions u1 (x), u2 (x), …, to yn (x). Functions u1 (x), u2 (x), …, yn (x) are called linearly independent on (a, b) if a linear combination k1u1 (x) +k2 u2 (x)+ … +knyn (x) =0, only at k1=k2= … =kn=0.
4. Now it is necessary to consider a question of justification of linear independence of a system of functions u1 (x), u2 (x), …, yn (x). Let them have derivatives to (n-1) - go about inclusive. The determinant made of these functions and their derivatives is called Vronsky's determinant (see fig. 2) or vronskniany.
5. Creation of determinant of Vronsky made of solutions of LODU L [y] =0 on an interval (a, b), allows to answer a question of whether these decisions linearly - are dependent. It is simple to prove that if functions u1 (x), u2 (x), …, yn (x) are linearly dependent on an interval (a, b), Vronsky's determinant of these functions is equal to zero in all points of an interval. Considering the LODU this property, it is possible to formulate the following statement easily.
6. In order that solutions of LODU u1 (x), u2 (x), …, yn (x) with continuous on an interval (a, b) coefficients were linearly independent, is necessary also enough that their determinant of Vronsky of W (x) did not equal to zero in one point of this interval (a, b).
7. Only now, on a final step to give the final answer to the question posed. Any set of n of linearly independent private solutions of the equation (1) is called the fundamental system of decisions (FSD) of this equation. Besides, it becomes clear that the direct answer "as find" can be received by means of Vronsky's determinant only after the answer to the question "How to Solve LODU?".