Solve the differential equation given by x dy - y dx - √(x² + y²) dx = 0.

MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2023: question 28(b)

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, section C, question 28(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(b) · 3 marks · Short answerDifferential Equations

Solve the differential equation given by $x d y - y d x - x^{2} + y^{2} d x = 0$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

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

Hint 1

Divide through by $x d x$ and check whether the equation is homogeneous.

Hint 2

Substitute $y = v x$ . Recall $\int \frac{d v}{1 + v ^{2}} = lo g ∣ v + 1 + v^{2} ∣$ .

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.

Rewrite the equation as $\frac{d y}{d x} = \frac{y + x ^{2} + y ^{2}}{x} = \frac{y}{x} + 1 + (\frac{y}{x})^{2}$ ... (i). This is a homogeneous equation (taking $x > 0$ ).
½ mark
Put $y = v x$ , so $\frac{d y}{d x} = v + x \frac{d v}{d x}$ . Substituting in (i): $v + x \frac{d v}{d x} = v + 1 + v^{2}$ .
½ mark
Separating variables: $\frac{d v}{1 + v ^{2}} = \frac{d x}{x}$ .
½ mark
Integrating both sides: $lo g v + 1 + v^{2} = lo g ∣ x ∣ + lo g C$ .
1 mark
So $\frac{y}{x} + \frac{x ^{2} + y ^{2}}{x} = C x$ , which gives $y + x^{2} + y^{2} = C x^{2}$ .
½ mark

Answer: $y + x^{2} + y^{2} = C x^{2}$

Where the marks are

Expresses the equation in homogeneous form (½ mark)
Substitutes $y = v x$ (½ mark)
Separates the variables (½ mark)
Integrates correctly (1 mark)
Final general solution $y + x^{2} + y^{2} = C x^{2}$ (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 paper2023 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 28(b)

Hints

Step-by-step solution

Revise the chapter

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