General properties of solutions. Continuity, differentiation of solutions of parameter axis and initial data.

Consider the IVP with a parameter s ∈ Rm

(1)

where f: Ω → Rn and Ω is an open subset of . Here the triple (t, x, s) is identified as a point in follows:

(t, x, s) = (t, x1,.., xn, s1,..., sm).

For any s ∈ , consider the open set

Ωs ={(t, x) ∈ : (t, x, s) ∈ Ω} Denote by S the set of those s, for which Ωs contains (t0, x0), that is,

S = {s ∈ : (t0, x0) ∈ Ωs}

= {s ∈ : (t0, x0, s) ∈ Ω}

Then the IVP can be considered in the domain Ωs for any s ∈ S. We always assume that the set S is non-empty. Assume also in the sequel that f (t, x, s) is a continuous function in (t, x, s) ∈ Ω and is locally Lipschitz in x for any s ∈ S. For any s ∈ S,

denote by x (t, s) the maximal solution of (1) and let Is be its domain (that is, Is is an

open interval on the axis t). Hence, x (t, s) as a function of (t, s) is defined in the set

U ={(t, s) ∈ : s ∈ S, t ∈ Is}

Theorem 1. Under the above assumptions, the set U is an open subset of a nd

the function x (t, s): U → is continuous.

Differentiability Of solutions.

A property of solutions of differential equations, that the solutions posses a specific number of continuous derivatives with respect to the independent variable and the parameter appearing in the equation.

Consider an equation of the type (x may also be a vector):

= f(t, x, μ)(1)

(1)

where μ is a parameter (usually also a vector), and let x(t, μ) be a solution of (1) defined by the initial condition

x| t=t0=x0 (2) First differentiability of the solution with respect to t is considered. If f is continuous with respect to t and x, the theorem on the existence of a continuous solution of the problem (1)–(2) is applicable in some domain, and then it follows from the identity which is obtained after substitution of x(t, μ) in (1) that the continuous derivative xt also exists. The presence of n continuous derivatives of f with respect to t and x means that there exist n+1 continuous derivatives of the solution with respect to t; xt^(n) may be found (expressed in terms of x(t, μ)) by successive differentiation of the identity obtained by substituting x(t, μ) in (1).

Linear equations of the n-th order. Basic properties.

Let’s consider the differential equation

=f(x ) (1)

which coefficients p 1(x),..., pn (x) and the constant term f (x) are continuous on the interval J ⊆ (− ∞,∞). It is required to find a solution of equation (1) with conditions

y(x0)=y0, y’(x0)= , …. (2)

The problem (1), (2) is called Cauchy problem.

Theorem 1. Let the coefficients pi (x), i = 1, n of equation (1) and the right side f (x) be continuous on the interval J.Then it has a unique solution y = ϕ(x)∈ (J) satisfying to the initial conditions (2). We introduce a special notation for the left side of equation (1) L (y) ≡ + p (x) pn (x) y (3)

and we call L as n order linear differential operator (acting in (J)).

Concept of operator generalizes the notion of function. In this situation, the operator L for each function y ∈ (J) assigns a function L (y)∈ V (J) by the formula (3). Property of the operator L(c 1 y 1 + c 2 y 2)= c 1 L(y 1) + c 2 L(y 2), (4)

is called a linearity of operator.If in equation (1) f (x) vanishes, we obtain the equation L (y) = 0, (5) which is called homogeneous. And the equation (1) is called a linear inhomogeneous. Further, we consider the homogeneous equation (5).

Theorem 2. Linear combination solutions of equation (5) is a solution.

10) Linear homogeneous equations, Solution properties. the Euler method which is a method of constructing a fundamental system of solutions for the equation is

(1)

Following Euler we find solutions of equation (1) in the form y = , where λ is a constant. We prove that equation (1) has a solution of the form y = , if λ satisfies to the equation l( λ ) = 0, (2) where l( λ )=

Equation (2) is called characteristic for the differential equation (1).

It is considered separately the different cases.

1) The roots of the characteristic polynomial l( λ ) are real and different.

2) The roots of the characteristic polynomial are different, but among of them there are complex roots.

3) The roots of the characteristic equation are real, but among of them there are multiple. To multiple root λ =λ * multiplicity " k " corresponding to " k " linearly independent solutions of the form y 1= , y 2=x , … =

4) In the general case can also be multiple complex roots. Note that if the multiplicity of the root α + i β is e, that multiplicity of the adjoint root as well e. Therefore, such pair of roots corresponding to the following linearly independent solutions

e ^{α x} cosβx, xe ^{α x} cosβx, …….., x ^{l -1} e ^{α x} cosβx

e ^{α x} sinβx, xe ^{α x} sinβx, …….., x ^{l -1} e ^{α x} sinβx