A. $\frac{1}{f\left(D^{2}\right)} \sin ax = \frac{\sin ax}{f\left(-a^{2}\right)}$
B. If $f\left(-a^{2}\right) = 0$, then $\frac{1}{f\left(D^{2}\right)} \sin ax = x \frac{\sin ax}{f'\left(-a^{2}\right)}$
C. If $f\left(-a^{2}\right) = 0$, then $\frac{1}{f\left(D^{2}\right)} \sin ax = x \frac{\cos ax}{f'\left(-a^{2}\right)}$
D. If $f'\left(-a^{2}\right) = 0$, then $\frac{1}{f\left(D^{2}\right)} \sin ax = x^{2} \frac{\sin ax}{f''\left(-a^{2}\right)}$
Choose the correct answer from the options given below :
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.
Important Questions from Mixed Topic (CUET PG)
- Who was the founder of Bolshevik Communist party?
- What is the key guide to statecraft in the realist tradition?
- Chronologically arrange the events in the Cold War period.
A. Berlin Wall is constructed
B. Communist China joins the UN
C. Soviet invasion of Czechoslovakia
D. Berlin Blockade
Choose the correct answer from the options given below: Morgenthau's principles of political realism are:
A. Politics is rooted in permanent and unchanging human nature which is basically self centred, self-regarding and self-interested
B. Politics is an autonomous sphere of action and cannot therefore be reduced to morals
C. International Politics is an arena of conflicting self-interests
D. The ethics of international relations is situational ethics which is very different from private morality
Choose the correct answer from the options given below:Who among the following political thinkers consider the anarchical self help system to be a compelling factor for States to maximise their relative power positions?
