- •Basic concepts and definitions of differential equations.
- •Equations with separated variables.
- •Linear equations of the first order.
- •Equations of the first order unsolved by derivatives.
- •6) Method of introducing a parameter. Lagrange and Clairaut equations.
- •7) Theorem about existence and uniqueness of Cauchy problem.
- •8) General properties of solutions. Continuity, differentiation of solutions of parameter axis and initial data.
- •9) Linear equations of the n-th order. Basic properties.
- •11) Linear inhomogeneous equations
- •12) Method of variation of arbitrary constants.
- •13) Integration of linear equations with permanent coefficients.
- •14) Euler method.
- •16) Fundamental system of solutions.
- •18) Integration of linear inhomogeneous system with quasi-polynomial right side.
- •20.Boundary problem for system of the second order. Green function.
- •Local properties of solutions.
- •Global properties of solutions.
- •26) Transformation of solutions and system.
- •27. Basic concepts. Stability by Lyapunov. Geometric means.
- •28. The main Lyapunov theorems.
- •30) Studying stability by Lyapunov function.
- •31.Differential equations in full differentials .Integral multiplier.
- •32.Integration methods. Homogeneous and linear differential equations of the first order.
- •33.Incomplete equations . Equations assuming reduction of order.
- •34) Local and global theorems.
- •35) Analyticity and differentiation of solutions.
- •36) Solution continuity of parameters and initial data.
- •37. Fundamental system of solutions
- •38. Wronskian determinant. Liouville formula.
- •39. Homogeneous and inhomogeneous linear equations. Liouville formula.
- •40 Integration of linear inhomogeneous equation with quasipolinomial right side.
- •41) Cauchy function. Fundamental solutions.
- •42. Systems of linear differential equations with constant coefficients
- •43. Linear systems of differential equations. Basic systems
- •44. Some methods of the system integration (leading to one equation etc)
- •45.Method of variation of constants
44. Some methods of the system integration (leading to one equation etc)
We know some methods of the system integration: Euler method of integration of homogeneous linear differential equations with constant coefficients, Method of variation of constants, Method of undetermined coefficients etc. We will discuss in detail these 3 more basic methods.We consider a linear system of differential equations:
(1)
where fi(t)-continuous functions in an interval;the coefficients aij(i, j = 1,2,…,n)- constants. The easiest way to integrate a system by reducing it to one equation of higher order, and this would also be a linear equation with constant coefficients. We write the system in matrix form:
(2)
The general solution of (2) has the structure: X=Xg.s+Xp.s, where Xg.sthe general solution of the homogeneous system
(If the vector f(t) is identically equal to zero: f(t)=0, then the system is said to be homogeneous)
We consider Euler method of integration of homogeneous linear differentialequations with constant coefficients.
According to this method, we find the solution of (4) as:
X = Beλt, (5)
Where B= is unknown column vector, λis unknown number.
Substituting (5) into (4), we obtain the matrix equation:AB=λB, (B ) (6)
λ (λ ≠ 0) is the eigenvalue of the matrix A; vector B is eigenvector corresponding to λ, is found from det(A −λE) = 0, (7))
whereE is the identity matrix n × n .
For a given eigenvalue λcomponents corresponding eigenvector B is found of the system of linear homogeneous equations:
Under solving of the system the following cases is possible:
1. All the roots of (7) are real and different.
2. The roots of (7) are real, but some of them are multiple.
3. The characteristic equation (7) has a complex variety of roots
According to the theory, in order to make a valid decision, it is enough to take the real and imaginary parts of one of the solutions:X1= ReY1, X2= ImY1.
Constructed in this way, the system solutions, the so-called fundamental system of solutions would consist of real functions.
Among the roots of the characteristic equation there are multiple complex roots. In this case the solution is to be found by analogy with the case of 2. Then separating the real and imaginary parts, we obtain so-called fundamental system of
(linearly independent) solutions of real functions.
Let us, finally, to find a particular solution Xp.sof (1) has the right part of a special type, if the vector F is:
F = (P(t)cosbt+Q(t)sinbt)eat , where P(t),Q(t) - vector polynomials of the form m H0+ H1 t + H2 t +…+ Hmtm
(not necessarily the same order) with vector coefficients H0,H1,H2,…,Hm. For linear systems with right-special form a particular solution can be found in the same form as the right-hand side, but with unknown coefficients. These coefficients are determined by substituting the source solution to the equation and equating like terms in the left and right sides of the matrix equation. The above method is known as the method of undetermined coefficients.