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` .