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`.