List of basic formulas and concepts related to higher-order linear differential equations:
- 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. - Homogeneous equation:
[
a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \ldots + a_1(x) y’ + a_0(x) y = 0
] - 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. - 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. - 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. - Linear operators:
A linear differential equation can be written as:
[
L[y] = f(x)
]
where ( L ) is a linear differential operator.