Find the particular solution of the differential equation (x²+y²)dy=xy dx, given that y(0)=1

MathsCBSE Class 12 · Sample paper, session 2026-27 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2026-27: question 29(B)

From the official CBSE sample question paper for 2026-27, 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 $(x^{2} + y^{2}) d y = x y d x$ , given that $y (0) = 1$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

The right side $\frac{x y}{x ^{2} + y ^{2}}$ is homogeneous of degree 0.

Hint 2

Put $y = v x$ , separate the variables, and split $\frac{1 + v ^{2}}{v ^{3}}$ into $\frac{1}{v ^{3}} + \frac{1}{v}$ .

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.

$\frac{d y}{d x} = \frac{x y}{x ^{2} + y ^{2}}$ is homogeneous. Put $y = v x$ , so $\frac{d y}{d x} = v + x \frac{d v}{d x}$ . Then $v + x \frac{d v}{d x} = \frac{v}{1 + v ^{2}} \Rightarrow x \frac{d v}{d x} = \frac{v - v - v ^{3}}{1 + v ^{2}} = \frac{- v ^{3}}{1 + v ^{2}}$
1 mark
Separating variables: $\int (\frac{1}{v ^{3}} + \frac{1}{v}) d v = - \int \frac{d x}{x} \Rightarrow - \frac{1}{2 v ^{2}} + lo g ∣ v ∣ = - lo g ∣ x ∣ + lo g C$ . So $lo g \frac{v x}{C} = \frac{1}{2 v ^{2}}$ , i.e. $lo g \frac{y}{C} = \frac{x ^{2}}{2 y ^{2}}$ , or $y = C e^{\frac{x ^{2}}{2 y ^{2}}}$
1 mark
Using $y (0) = 1$ : $1 = C e^{0} \Rightarrow C = 1$ . The particular solution is $y = e^{\frac{x ^{2}}{2 y ^{2}}}$
1 mark

Answer: $y = e^{\frac{x ^{2}}{2 y ^{2}}}$

Where the marks are

Substitutes $y = v x$ and reduces to $x \frac{d v}{d x} = \frac{- v ^{3}}{1 + v ^{2}}$ (1 mark)
Separates, integrates and gets the general solution $y = C e^{x^{2} /2 y^{2}}$ (1 mark)
Uses $y (0) = 1$ to get $C = 1$ and the particular solution (1 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 paperSample paper 2026-27: every question, by section.

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CBSE Class 12 Mathematics sample paper 2026-27: question 29(B)

Hints

Step-by-step solution

Revise the chapter

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