12) Method of variation of arbitrary constants.

The method of variation of parameters with arbitrary constants (Lagrange's method) is used to construct the general solution of the nonhomogeneous equation, when we know the general solution of the associated homogeneous equation.  Suppose that the general solution of the second order homogeneous equation is expressed through the fundamental system of solutions  y₁(x) and y₂(x): y₀(x)=C₁y₁(x)+ C₂y₂(x) where C₁, C₂ are arbitrary constants. The idea of this method is to replace the constants C₁ and C₂ by functions C₁(x) andC₂(x), which are chosen so that the solution satisfies the nonhomogeneous equation.  The derivatives of the unknown functions C₁(x) and C₂(x) can be determined from the system of equations

The main determinant of this system is the Wronskian of the functions y₁ and y₂, which is not equal to zero due to linear independence of the solutions y₁ and y₂. Therefore, this system of equations always has a unique solution. The final formulas for C₁' (x) and C₂' (x) have the form

C₁`(x)= ; C<sub>2</sub>`(x)=

When using the method of variation of parameters, it is important to remember that the function f(x) must correspond tothe differential equation in the standard form, i.e. the coefficient a₀(x) at the second derivative must be equal to 1.  Then the general solution of the original nonhomogeneous equation will be expressed by the formula y(x)= C₁(x)y₁(x)+ C₂(x)y₂(x) =

[ - dx+A₁ ] y1(x) + [ - dx+A₂ ]y2(x) =

A ₁y₁(x)+A ₂y₂(x)+Y(x)

Where Y(x) denotes a particular solution of the nonhomogeneous equation. 

13) Integration of linear equations with permanent coefficients.

Euler method which is a method of constructing a fundamental system of solutions for the equation

a₀y⁽ⁿ⁾+a₁y^(n-1)+…+a_ny=0 (1)

Following Euler we find solutions of equation (1) in the form y = e^λx , where λ is a constant.

We prove that equation (1) has a solution of the form y = e^λx , if λ satisfies to the equation l(λ ) = 0 , (2) where l(λ ) = λⁿ+a₁λⁿ⁻¹+ ...+ a_n

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

y₁=e ^λ*x_, y₂=xe ^λ* ,..,y_k=x^k-1e ^λ*

4) In the general case can also be multiple complex roots. Note that if the multiplicity of the root α + iβ is e , that multiplicity of the adjoint root as well e . Therefore, such pair of roots

corresponding to the following linearly independent solutions

e^αxcosβx, xe^αxcosβx,...,x^l-1e^αxcosβ x,

e^αxsinβx, xe^αxsinβx,...,x^l-1e^αxsinβ x,

The method of undetermined coefficients

This method of selection of a particular solution for inhomogeneous linear equation with constant coefficients, when the in-homogeneity of quasi-polynomial. Quasi-polynomial are called functions of the form f(x)=

where μ_j are constants (not necessarily real), and Pj ( x ) are polynomials of x .

For selecting of particular solutions, the following theorem is used.

Theorem 9. Equation a₀y⁽ⁿ⁾+a₁y^(n-1)+…+a_ny=P(x)e^μx

where P ( x ) is a polynomial of degree l , has a particular solution of the form Y_r(x)=x^sQ(x) e^μx

where s is equal to zero, if μ is not the root of the characteristic equation of the operator L , otherwise s is equal to the multiplicity of the root μ , and Q( x ) is polynomial of the same

degree l as P ( x ).