List of basic formulas and concepts related to higher-order linear differential equations:

  1. The general equation of a linear differential equation of the n-th order is:
    [
    a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \ldots + a_1(x) y’ + a_0(x) y = f(x)
    ]
    where ( a_n(x), a_{n-1}(x), \ldots, a_0(x) ) are functions depending on ( x ), and ( f(x) ) is a given function.
  2. Homogeneous equation:
    [
    a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \ldots + a_1(x) y’ + a_0(x) y = 0
    ]
  3. Solution of a homogeneous equation:
    If ( y_1, y_2, \ldots, y_n ) are linearly independent solutions of a homogeneous equation, then the general solution can be written as:
    [
    y_h = C_1 y_1 + C_2 y_2 + \ldots + C_n y_n
    ]
    where ( C_1, C_2, \ldots, C_n ) are arbitrary constants.
  4. Method of variation of arbitrary constants:
    To find a particular solution ( y_p ) of a non-homogeneous equation, one can use:
    [
    y_p = C_1(x) y_1 + C_2(x) y_2 + \ldots + C_n(x) y_n
    ]
    where ( C_i(x) ) are functions that are determined by substitution into the equation.
  5. Wronski’s formula:
    To check the linear independence of solutions ( y_1, y_2, \ldots, y_n ):
    [
    W(y_1, y_2, \ldots, y_n) = \begin{vmatrix}
    y_1 & y_2 & \ldots & y_n \
    y_1′ & y_2′ & \ldots & y_n’ \
    \vdots & \vdots & \ddots & \vdots \
    y_1^{(n-1)} & y_2^{(n-1)} & \ldots & y_n^{(n-1)}
    \end{vmatrix}
    ]
    If ( W \neq 0 ), then the solutions are linearly independent.
  6. Linear operators:
    A linear differential equation can be written as:
    [
    L[y] = f(x)
    ]
    where ( L ) is a linear differential operator.

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