6. Reduction of Order.

There are 3 main types of equations that allow the reduction of order. (1) the method for integrating the right side of equation Consider DE’s form as: y⁽ⁿ⁾=f(x). It is solved by successive integration the right side, and it will be integrate n times.

_{Ex: y’’=x}²-2x

(2) DE explicitly missing the function y Ex: replace y’’=z, y’’’=z’ thus, we’ve reduced the equation to the first order. Now share variables and integrate: Perform inverse replacement: z=y’’ and it will be the general solution, where С_1,2,3=const

(3) In DE the independent variable x is missed: Ex: (y-1)y’’=2(y’)², y(0)=2, y’(0)=2 Replace y’=z(y) and find the second derivative: y’’=(z(y))’=z’(y)y’=z’z and replace the original equation:

Carry out the inverse replacement:

_y’=C1(y-1)² and use both the initial conditions: y(0)=2, y’(0)=2 2= C₁(2-1)² и C₁=2, thus y’=2(y-1)² and the continuous is simple: using initial conditions:

6. Linear homogeneous and non-homogeneous de of n-th order

(3) – characteristic equation Roots: 1) real&distinct (RD) , , , 2)complex&distinct (CD) , , , , 3)real&multiple (RM) - order of equation 4)complex&multiple compose the characteristic equation, and find its root: , ,

_{Linear Inhomogeneous DE of n}^th order with const coefficients method of Undetermined coefficients _{, , 1) α – is not root of Ch. Eq. Ly1=f1, Ly2=f2, Qk(x)eαx 2) α – root of CE, S – multiple Qk+s(x) eαx}

_{Inhomogeneous DE , p=const, , 1) n=1, y’+p1y=f(x) z-solution of homogeneous DE y=C(x)z, y’=C’(x)z+C(x)z’ C’(x)z+C(x)z’+ p1 C(x)z=f(x) C’(x)z+C(x)[ z’+ p1z]=f(x) C’(x)z=f(x) 2) n=2, y’’+p1y’+p2y=f(x), z1,z1 – solution of HDE} { z₁,…,z_n – FSS, y=c₁z₁+…+c_nz_n P_n-1→ y=c₁z₁+…+c_nz_n

6. The theorem of existence and uniqueness of the solution for Linear de

(1)

1. (2)

2. ; on R

Lipschutz’ solution.

Successful approximation.

solution of (existence)

Uniqueness of solution.

Systems of DE of first order

14. Linear independent system of functions.

15. Wronskian.

Definition: System of functions y₁(x), y₂(x) is called linear independent system of functions on (a,b) if there don’t exist such if if it’s determinant equals 0, it's linear dependent.

The Wronskian Theorems

Suppose that

is an n-th order linear differential operators with continuous real valued coef-

ficients on an interval I. Suppose that for j = 1, 2,... ,n, t) satisfies L, = 0.

Definition. The wronskian of these n solutions is defined as the n × n determinant,

W(t):= Definition. Theo lutions are linearly dependent when there are constants not all zero so that Otherwise

they are linearly independent.

General Wronskian Theorem. The following are equaivalent.

1. For some

2. For any values there are uniquely determined constants so that y = is the solution of the initial value problem,

Ly = 0, y( ) = , y′( ) = , … ( ) = .

3. Every solution of Ly = 0 is of the form for real constants . That is, is the general solution.

4. 4. For all t ∈ I, W(t) 0.

5. The solutions y1,y2,... ,yn are linearly independent.

Example:

Show that the functions x, sin x, cos x are linearly independent.

Solution.

We find the Wronskian matrix W(x) for the system of functions:

      

Since the Wronskian is not identically zero, it follows that this system of functions is linearly independent.