Linear Differential Equations of Arbitrary Order

A linear differential equation of order \( k \) is one that involves a linear combination of the function \( y(x) \) and its derivatives up to order \( k \): $$ a_k(x)y^{(k)}(x) + \dots + a_1(x)y'(x) + a_0(x)y(x) = f(x) $$ Here, \( a_k(x), \dots, a_0(x) \) are the coefficient functions, while \( k \) denotes the highest derivative appearing in the equation.

If \( f(x) = 0 \), the equation is called homogeneous.

If \( f(x) \neq 0 \), it is referred to as nonhomogeneous.

Note: A linear differential equation can be written in standard form, provided that the leading coefficient \( a_k(x) \) is nonzero.

The category of linear differential equations encompasses a wide range of subtypes and forms.

For instance, a first-order linear differential equation with variable coefficients has the form:

$$ y'(x) + a(x)y(x) = b(x) $$

Meanwhile, linear equations of arbitrary order with constant coefficients take the form:

$$ a_k y^{(k)}(x) + \dots + a_1 y'(x) + a_0 y(x) = f(x) $$

In this case, the coefficients are constants - they do not depend on the independent variable \( x \).

Other cases follow similar principles.