# 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=bar(0..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, the differential equation is called homogeneous; otherwise, it is called 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 are not constant, the 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=bar(0..n)) are continuous in some interval I containing x_0 and if b_n(x)!=0 in I, the given differential equation together with the initial conditions has a unique (only one) solution defined throughout I.

When the conditions on b_n(x) in the theorem hold, we can divide the 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=bar(0..n-1)) and phi(x)=(g(x))/(phi(x)).

Now, let's define the 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, a differential equation can be rewritten as L(y)=phi(x), and, in particular, a linear homogeneous differential equation can be expressed as L(y)=0.

A 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, we have that 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, we have that L(c_1 y_1+c_2y_2+...+c_n y_n)=0 for any constants c_i (i=bar(1..n)).

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