The integrating factor of differential equation (x+2y³)dy/dx=2y is (A) e^(y²)/(2) (B) (1)/(√(y)) (C) (1)/(y²) (D) e^-(1)/(y²)

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 1 mark · MCQ

CBSE Class 12 Mathematics 2025: question 12

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section A, question 12, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q12 · 1 mark · MCQDifferential Equations

The integrating factor of differential equation $(x + 2 y^{3}) \frac{d y}{d x} = 2 y$ is

(A) $e^{\frac{y ^{2}}{2}}$
(B) $\frac{1}{y}$
(C) $\frac{1}{y ^{2}}$
(D) $e^{- \frac{1}{y ^{2}}}$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Try writing the equation as $\frac{d x}{d y} + P x = Q$ .

Hint 2

The integrating factor is $e^{\int P d y}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Treat $x$ as the dependent variable: $\frac{d x}{d y} = \frac{x + 2 y ^{3}}{2 y} \Rightarrow \frac{d x}{d y} - \frac{1}{2 y} x = y^{2}$ .
I.F. $= e^{- \int \frac{1}{2 y} d y} = e^{- \frac{1}{2} l o g y} = \frac{1}{y}$ , option (B).
1 mark

Answer: $\frac{1}{y}$ , option (B)

Where the marks are

Writes the equation in linear form in $x$ and finds the I.F. (1 mark)

Revise the chapter

Chapter: Differential Equations (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Differential Equations, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 12

Hints

Step-by-step solution

Revise the chapter

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