# Basic Concepts

An nth-order linear differential equation has the form b_n(x)y^((n))+b_(n-1)(x)y^((n-1))+...+b_2(x)y''+b_1(x)y'+b_0(x)y=g(x) where g(x) and all coefficients b_j(x) (j=0 ,1 ,2 ,..., n) depend solely on the variable x. In other words. they do not depend on y or on any derivative of y.

If g(x)-=0 then differential equation is called homogeneous, otherwise, non-homogeneous. For example x^3 y'''+1/xy''+2y'=0 is homogeneous and y''+xy'=cos(x) is non-homogeneous.

A linear differential equation has constant coefficients if all the coefficients b_j(x) are constants: if one or more of these coefficients is not constant then differential equation has variable coefficients. For example, y^((4))+x^2 y'+xy=6 has variable coefficients and 3y''+y'=e^x has constant coefficients.

Theorem. Consider the initial-value problem given by the linear differential equation and the n initial conditions y(x_0)=y_0 , y'(x_0)=y_0' , y''(x_0)=y_0'' , ..., y^((n-1))(x_0)=y_0^(n-1) .

If g(x) and b_j(x) (j=0, 1, 2, ..., n) are continuous in some interval I containing x_0 and if b_n(x)!=0 in I, then the given differential equation together with initial conditions has unique (only one) solution defined throughout I.

When conditions on b_n(x) in theorem hold, we can divide differential equation by b_n(x) to obtain y^((n))+a_(n-1)(x)y^((n-1))+...+a_2(x)y''+a_1(x)y'+a_0(x)y=phi(x) where a_j(x)=(a_j(x))/(b_n(x)) (j=0, 1, ..., n-1) and phi(x)=(g(x))/(phi(x)) .

Now, let's define differential operator L(y) by L(y)=y^((n))+a_(n-1)(x)y^((n-1))+...+a_2(x)y''+a_1(x)y'+a_0(x)y then differential equation can be rewritten as L(y)=phi(x) and, in particular, a linear homogeneous differential equation can he expressed as L(y)=0 .

Differential operator has two properties:

If L(y)=0 then L(cy)=0 for any constant c.

If L(y_1)=0 and L(y_2)=0 then L(y_1+y_2)=0 .

These two properties can be combined into one property: if L(y_1)=0, L(y_2)=0, ..., L(y_n)=0 then L(c_1 y_1+c_2y_2+...+c_n y_n)=0 for any constants c_i (i=1, 2, ..., n).

What does it gives us? It gives us the following fact: if we have n solutions y_1, y_2, ..., y_n that satisfy given homogeneous differential equation, so their linear combination will also satisfy same homogeneous differential equation. It is only question what form should have y_1, y_2, ..., y_n so their linear combination will form general solution of homogeneous differential equation?

Example. y_1=cos(t) and y_2=sin(t) are solutions of differential equation y''+y=0 . So, y_g=c_1y_1+c_2y_2=c_1cos(t)+c_2sin(t) is also solution for any constants c_1 and c_2 .