The integrating factor of the differential equation (1-x²)dy/dx + xy = ax, -1 < x < 1, is : (A) (1)/(x²-1) (B) (1)/(√(x²-1)) (C) (1)/(1-x²) (D) (1)/(√(1-x²))

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 1 mark · MCQ

CBSE Class 12 Mathematics 2024: question 8

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section A, question 8, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q8 · 1 mark · MCQDifferential Equations

The integrating factor of the differential equation $(1 - x^{2}) \frac{d y}{d x} + x y = a x$ , $- 1 < x < 1$ , is :

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

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

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

Hint 1

First write the equation in the form $\frac{d y}{d x} + P y = Q$ .

Hint 2

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

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.

Dividing by $1 - x^{2}$ gives the linear form $\frac{d y}{d x} + \frac{x}{1 - x ^{2}} y = \frac{a x}{1 - x ^{2}}$ , so $P = \frac{x}{1 - x ^{2}}$ .
½ mark
$\int P d x = - \frac{1}{2} lo g (1 - x^{2})$ , so $IF = e^{- \frac{1}{2} l o g (1 - x^{2})} = \frac{1}{1 - x ^{2}}$ .
½ mark

Answer: (D) $\frac{1}{1 - x ^{2}}$

Where the marks are

Finds $P = \frac{x}{1 - x ^{2}}$ and $IF = \frac{1}{1 - x ^{2}}$ (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 paper2024 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2024: question 8

Hints

Step-by-step solution

Revise the chapter

More questions from Differential Equations