MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2026: question 29(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section C, question 29(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(a) · 3 marks · Short answerDifferential Equations

Find the general solution of the following differential equation :
$x^{2} \frac{d y}{d x} = x^{2} + x y + y^{2}$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/1/1.

Hints

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

Hint 1

Divide by $x^{2}$ and notice the right side depends only on $\frac{y}{x}$ .

Hint 2

Use the substitution $y = v x$ .

Hint 3

After separating variables, recall $\int \frac{d v}{1 + v ^{2}} = tan^{- 1} v$ .

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 $x^{2}$ : $\frac{d y}{d x} = 1 + \frac{y}{x} + (\frac{y}{x})^{2}$ , a homogeneous equation.
½ mark
Put $y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$
½ mark
$v + x \frac{d v}{d x} = 1 + v + v^{2} \Rightarrow x \frac{d v}{d x} = 1 + v^{2} \Rightarrow \frac{d v}{1 + v ^{2}} = \frac{d x}{x}$
½ mark
Integrating: $\int \frac{d v}{1 + v ^{2}} = \int \frac{d x}{x} \Rightarrow tan^{- 1} v = lo g ∣ x ∣ + C$
1 mark
Replacing $v = \frac{y}{x}$ : $tan^{- 1} (\frac{y}{x}) = lo g ∣ x ∣ + C$
½ mark

Answer: $tan^{- 1} (\frac{y}{x}) = lo g ∣ x ∣ + C$

Where the marks are

Writes equation in homogeneous form (½ mark)
Substitutes $y = v x$ (½ mark)
Separates variables $\frac{d v}{1 + v ^{2}} = \frac{d x}{x}$ (½ mark)
Integrates to $tan^{- 1} v = lo g ∣ x ∣ + C$ (1 mark)
Back-substitutes $v = \frac{y}{x}$ (½ mark)

Revise the chapter

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

Practise this chapterBoard-style questions from Differential Equations, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 29(a)

Hints

Step-by-step solution

Revise the chapter

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