Find the particular solution of the differential equation xydy/dx = (x+2)(y+2), given that y(1) = -1.

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

CBSE Class 12 Mathematics 2026: question 29(b)

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

Q29(b) · 3 marks · Short answerDifferential Equations

Find the particular solution of the differential equation
$x y \frac{d y}{d x} = (x + 2) (y + 2)$ , given that $y (1) = - 1$ .

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

Bring all y terms with dy and x terms with dx.

Hint 2

Write $\frac{y}{y + 2} = 1 - \frac{2}{y + 2}$ and $\frac{x + 2}{x} = 1 + \frac{2}{x}$ .

Hint 3

Substitute $x = 1$ , $y = - 1$ to find the constant.

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.

Separating variables: $\frac{y}{y + 2} d y = \frac{x + 2}{x} d x \Rightarrow \int (1 - \frac{2}{y + 2}) d y = \int (1 + \frac{2}{x}) d x$
1 mark
$y - 2 lo g ∣ y + 2∣ = x + 2 lo g ∣ x ∣ + C$ (general solution)
1½ marks
Using $y (1) = - 1$ : $- 1 - 2 lo g 1 = 1 + 2 lo g 1 + C \Rightarrow C = - 2$ .
Particular solution: $y - 2 lo g ∣ y + 2∣ = x + 2 lo g ∣ x ∣ - 2$ , i.e. $y = x + 2 lo g ∣ x (y + 2) ∣ - 2$
½ mark

Answer: $y - 2 lo g ∣ y + 2∣ = x + 2 lo g ∣ x ∣ - 2$

Where the marks are

Separates variables correctly (1 mark)
Integrates to the general solution (1½ marks)
Finds $C = - 2$ and writes the particular solution (½ 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 paper2026 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 29(b)

Hints

Step-by-step solution

Revise the chapter

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