14) Euler method.

Given a complex- valued solution. It is proved that the real and imaginary parts of the complex - valued solution are solutions . the Euler method which is a method of constructing a fundamental system of solution for the equation

Following Euler we find solution of the equation in the form , where is a constant. We prove that equation has a solution of the form , if satisfies to the equation Where

Equation is called characteristic for . It is considered separately the different cases.

1. The roots of characteristic polynomial are real and different.

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

3. The roots of characteristic equation are real, but among of them there multiple.

In the 3^rd case a theorem is proved. To multiple roots multiplicity ‘”k” corresponding to “k” linearly independent solution of the form

4. In the general case can also be multiple complex roots. Note that is the multiplicity of the roots , that multiplicity of the adjoint root as well e. therefore, such pair of roots corresponding to the following linearly independent solutions

15) Basic properties of linear systems. Vector and matrix form.

Linear System Matrix-Vector Form: given an nxn matrix A(t) and nx1 vector valued function f(t):

= a_ik(t)x_k+f_i(t) i=1….n (pustoi kvadrattyn ornynda ewtene jok)

where a_ik(t),f_i(t) are given functions, x_i(t)is unknown functions called a linear system of differential equations.

We denote

Then (1) takes the form =A(t) + (t)

If f(t)=0, the system is homogeneous.if not-it’ll be inhomogeneous

Basic properties of linear system:

Homogeneous System Solution Properties :

-Linear combinations of solutions: if x1(t),x2(t),…xk()t,are solutions to x’=Ax,where x’=dx/dt, then x(t)=C₁x₁(t)+….+C_kx_k(t), is also solution for any constants C₁,…C_k .

-independence:suppose y1(t),y2(t),…y_k(t), are solutions to y’=Ay,for tє I=(a,b)

a)if y₁(t₀)..y_k(t_o) are dependent for some t₀ є I,then there is exist C1,C2…Ck,not all 0,so that C1y1(t)+C2y2(t)+….+C_k y_k (t)=0 t € I

(the y_i(t)'s are dependent any t є I );

b) if y₁(t₀)..y_k(t_o) are dependent for some t₀ є I,then y_i(t)'s are dependent any t є I ).

-Solution Structure: if y1(t)….yn(t),are linearly independent solutions to the n-dimensioanl system y’=Ay, then any solution has the form y(t)= C1y1(t)+C2y2(t)+….+C_ny_n (t) for some constants C₁,C₂….C_n.

The n y_i's form a fundamental set of solutions.

- Solution Strategy:

a)find n independent y_i's to form general solution;

b) if initial value y(t₀) is given,

solve Y(to)C=y(t0) for C=(C1,C2….Cn)^T, using nxn matrix we have Wronskian

y_11….. y_1n this expression called

y₂₁….. y_2n the wronskian

….

Y_n1….. y_nn