Any differential equation (DE), except required function and an argument comprises derivatives of this function. Differentiation and integration are the return operations. Therefore process of the decision (DU) often call it integration, and the decision – integral. Uncertain integrals contain any constants therefore DU also contain constants, and the decision defined to within constants is the general.
Instruction
1. The common decision of DU of any order could not be made absolutely. It is formed by itself if in the course of its receiving entry or regional conditions were not used. Another matter if a certain decision was not, and they got out on the set algorithms received on the basis of theoretical data. Quite so also occurs if it is about linear DU with constant in order n-go coefficients.
2. Linear uniform DU (LODU) n-go of an order has an appearance (see fig. 1). If to designate its left part as linear differential operator L [y], then LODU will correspond in the form of L [y]=0, and the L [y] =f(x) – for the linear non-uniform differential equation (LNUDE).
3. If to look for solutions of LODU in the form of y=exp(k∙x), then y '=k∙exp(k∙x), y’ '= (k^2) of ∙exp (k∙x), …, y^(n-1)= (k^(n-1)) ∙exp(k∙x), y^n=(k^n)∙exp(k∙x). After reduction on y=exp(k∙x), you will come to the equation: k^n+(a1)k^(n-1)+ … +a(n-1) ∙k+an=0, to the called characteristic. This usual algebraic equation. Thus, if k is a root of the characteristic equation, then the y=exp [kx] function is the solution of LODU.
4. The algebraic equation of n-y of degree has n of roots (taking into account multiple and complex). "One" corresponds to each material root ki of frequency rate the y=exp function [(ki)x] therefore if all of them valid and various, then taking into account that any linear combination of these the exhibitor is the decision too, it is possible to make the common decision of LODU: y=C1∙exp [(k1) of ∙x] + C2∙exp [(k2) of ∙x]+ … +Cn∙exp [(kn) of ∙x].
5. Generally, among solutions of the characteristic equation there can be material multiple and in a complex interfaced roots. At creation of the common decision in the designated situation be limited to LODU of the second order. Here receiving two roots of the characteristic equation is possible. Let it will be in a complex interfaced couple of k1=p+i∙q and k2=p-i∙q. Application the exhibitor with such indicators will give complex-valued functions at the initial equation with the valid coefficients. Therefore they will be transformed on Euler's formula and led to a type of y1=exp (p∙x) ∙sin(q∙x) and y2=exp(p∙x)cos(q∙x). For a case of one material root of frequency rate of r=2 use y1=exp(p∙x) and y2=x∙exp(p∙x).
6. Final algorithm. It is required to make the common decision of LODU of the second order of y’ ’+a1∙y ’+a2∙y=0. Work out the characteristic equation of k^2+a1∙k+a2=0. If it has the valid roots k1≠k2, then its common decision choose in the form of y=C1∙exp [(k1) of ∙x] + C2∙exp [(k2) of ∙x]. If there is one valid root k, frequency rates of r=2, then y=C1∙exp[k∙x] + C2∙x∙exp[k2∙x]=exp[k∙x] (C1+ C2∙x∙exp[k∙x]). If there is in a complex interfaced couple of roots k1=p+i∙q and k2=p-i∙q, then the answer write down in the form of y=C1∙exp(p∙x)sin(q∙x) ++ C2∙exp(p∙x)cos(q∙x).