Inverse Operator
The operator \( \frac{1}{D} \) is called the inverse operator of the differential operator \( D \). Thus:
\[ \frac{1}{D} \left[ Q(x) \right] = \int Q(x) \, dx \]
Particular Integral (P.I.) of \( f(D)y = Q(x) \)
Consider the differential equation:
\[ f(D)y = Q(x) \quad \text{(1)} \]
The expression \( \frac{1}{f(D)} Q(x) \) is a function of \( x \), not containing arbitrary constants, such that when operated upon by \( f(D) \), it gives \( Q(x) \). That is:
\[ f(D) \left[ \frac{1}{f(D)} Q(x) \right] = Q(x) \]
Thus, \( \frac{1}{f(D)} Q(x) \) satisfies equation (1) and is called the Particular Integral (P.I.) of (1).
Clearly, \( f(D) \) and \( \frac{1}{f(D)} \) are inverse operators.
Rules for Finding Particular Integral (P.I.)
Consider the differential equation:
\[ f(D)y = Q(x) \]
where:
\[ f(D) = D^n + a_1 D^{n-1} + a_2 D^{n-2} + \dots + a_{n-1} D + a_n \]