Find the particular solution of the differential equation given by x²dy/dx - xy = x²² (y/2x ), given that when x = 1, y = (π)/(2).

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 29

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section C, question 29, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29 · 3 marks · Short answerDifferential Equations

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

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

Divide by $x^{2}$ ; the right side depends only on $y / x$ .

Hint 2

Use $y = v x$ and separate variables.

Hint 3

$\int sec^{2} (v /2) d v = 2 tan (v /2)$ ; then use the initial condition.

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 $x^{2}$ : $\frac{d y}{d x} = \frac{y}{x} + cos^{2} (\frac{y}{2 x})$ , a homogeneous equation.
½ mark
Put $y = v x$ , so $\frac{d y}{d x} = v + x \frac{d v}{d x}$ .
½ mark
$v + x \frac{d v}{d x} = v + cos^{2} \frac{v}{2} \Rightarrow \int sec^{2} \frac{v}{2} d v = \int \frac{d x}{x}$
½ mark
$2 tan \frac{v}{2} = lo g ∣ x ∣ + C \Rightarrow 2 tan (\frac{y}{2 x}) = lo g ∣ x ∣ + C$
1 mark
At $x = 1, y = \frac{π}{2}$ : $2 tan \frac{π}{4} = lo g 1 + C \Rightarrow C = 2$ . Particular solution: $2 tan (\frac{y}{2 x}) = lo g ∣ x ∣ + 2$
½ mark

Answer: $2 tan (\frac{y}{2 x}) = lo g ∣ x ∣ + 2$

Where the marks are

Writes in form dy/dx = y/x + cos²(y/2x) (0.5 mark)
Substitutes y = vx (0.5 mark)
Separates variables (0.5 mark)
Integrates to get general solution (1 mark)
Finds C = 2 and particular solution (0.5 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 paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 29

Hints

Step-by-step solution

Revise the chapter

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