Class CBSE Class 12 Mathematics Differential Equations Q #1848
KNOWLEDGE BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
The integrating factor of differential equation $R\frac{dx}{dy} + Px = Q$ where P, Q, R are functions of y is
(A) $e^{\int\frac{P}{Q}dy}$
(B) $e^{\int Pdy}$
(C) $e^{\int\frac{P}{R}dy}$
(D) $e^{\int\frac{P}{R}dx}$
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Correct Answer: C

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Detailed Solution

Step 1: Standardize the Differential Equation

The given differential equation is R(dx/dy) + Px = Q. To find the integrating factor, we must first convert this into the standard linear form: dx/dy + P'(x) = Q'. Dividing the entire equation by R, we get: $$ \frac{dx}{dy} + \frac{P}{R}x = \frac{Q}{R} $$

Step 2: Identify the Coefficient of x

Comparing the standardized equation with the general form dx/dy + P_1(y)x = Q_1(y), we identify the coefficient of x as: $$ P_1(y) = \frac{P}{R} $$

Step 3: Calculate the Integrating Factor (IF)

The formula for the integrating factor of a linear differential equation in x is given by: $$ IF = e^{\int P_1(y) dy} $$ Substituting the value of P_1(y): $$ IF = e^{\int \frac{P}{R} dy} $$

Final Answer: Option (C)

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must manipulate the given equation into a standard form before applying the known formula for the integrating factor.
Knowledge Dimension: PROCEDURAL
Justification: The question tests the ability to execute a specific mathematical procedure (standardization of linear differential equations) to arrive at a solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly assesses the student's grasp of the standard methodology for solving linear differential equations as prescribed in the NCERT curriculum.

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