The question asks to identify the correct rules for finding the particular integral (PI) of a differential equation of the form $\frac{1}{f(D^2)} \sin ax$. We need to evaluate the given statements (A, B, C, D) based on standard methods for solving such differential equations.

Particular Integral Rules for \(\sin ax\)$

We analyze each statement:

• Statement A: The general rule for finding the particular integral $\frac{1}{f(D^2)} \sin ax$ is applicable when the denominator, evaluated at $D^2 = -a^2$, is non-zero. Specifically, if $f(-a^2) \neq 0$, then: $ \frac{1}{f(D^2)} \sin ax = \frac{\sin ax}{f(-a^2)} $ Statement A correctly states this rule. Therefore, A is correct.
• Statement B: This addresses the case where the denominator becomes zero, i.e., $f(-a^2) = 0$. When $f(-a^2) = 0$ but $f'(-a^2) \neq 0$, the rule is modified by differentiating the denominator with respect to $D$ (or effectively, $D^2$) and multiplying the numerator by $x$. The correct formula is: $ \frac{1}{f(D^2)} \sin ax = x \frac{\sin ax}{f'(-a^2)} $ Statement B correctly represents this rule. Therefore, B is correct.
• Statement C: This statement also considers the case $f(-a^2) = 0$ but suggests the result is $x \frac{\cos ax}{f'(-a^2)}$. This is incorrect. When starting with $\sin ax$ and applying the rule for $f(-a^2) = 0$, the resulting term should still involve $\sin ax$, not $\cos ax$. The correct form is given in Statement B. Therefore, C is incorrect.
• Statement D: This addresses a subsequent special case where not only $f(-a^2) = 0$ but also the first derivative of $f$ evaluated at $-a^2$ is zero, i.e., $f'(-a^2) = 0$. If $f''(-a^2) \neq 0$, the rule requires differentiating the denominator twice and multiplying the numerator by $x^2$: $ \frac{1}{f(D^2)} \sin ax = x^2 \frac{\sin ax}{f''(-a^2)} $ Statement D correctly states this rule. Therefore, D is correct.

Conclusion on Correct Statements

Based on the analysis, the correct statements are A, B, and D.

Matching with Options

We need to find the option that includes only A, B, and D.

• Option 1: A, B, and C only (Incorrect as C is wrong)
• Option 2: A, B, and D only (Correct)
• Option 3: A, C, and D only (Incorrect as C is wrong)
• Option 4: B, C, and D only (Incorrect as C is wrong)

The combination of correct statements A, B, and D matches Option 2.