General Form
The general form of an \( n \)-th order linear differential equation with constant coefficients is given by:
\[ \frac{d^n y}{dx^n} + a_1 \frac{d^{n-1} y}{dx^{n-1}} + a_2 \frac{d^{n-2} y}{dx^{n-2}} + \dots + a_{n-1} \frac{dy}{dx} + a_n y = Q(x) \]
where \( a_1, a_2, \dots, a_n \) are constants.
Differential Operator
Using the differential operator \( D = \frac{d}{dx} \), the equation can be written as:
\[ D^n y + a_1 D^{n-1} y + a_2 D^{n-2} y + \dots + a_{n-1} D y + a_n y = Q(x) \]
This can be expressed as:
\[ f(D) y = Q(x) \]
where:
\[ f(D) = D^n + a_1 D^{n-1} + a_2 D^{n-2} + \dots + a_{n-1} D + a_n \]
General (Complete) Solution
The general solution of \( f(D) y = Q(x) \) is given by:
\[ y = \text{C.F.} + \text{P.I.} = u(x) + v(x) \]
where:
- C.F. ( Complementary Function ) is the solution of the homogeneous equation \( f(D) y = 0 \).
- P.I. ( Particular Integral) is a specific solution of the non-homogeneous equation \( f(D) y = Q(x) \).
Homogeneous Differential Equation
If \( Q(x) = 0 \), the equation \( f(D) y = 0 \) is called a homogeneous differential equation, and its general solution is given by:
\[ y = \text{C.F.} \]