Solve the following differential equation : x² dy+y(x+y) dx=0

x²y=k(y+2x), k an arbitrary constant

MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 29(b)

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, 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

Solve the following differential equation : $x^{2} d y + y (x + y) d x = 0$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

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

Hint 1

Divide by $x^{2} d x$ ; the right side is a function of $\frac{y}{x}$ only.

Hint 2

Substitute $y = v x$ and separate variables.

Hint 3

Use $\int \frac{d u}{u ^{2} - a ^{2}} = \frac{1}{2 a} lo g \frac{u - a}{u + a}$ with $u = v + 1$ .

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.

$x^{2} d y = - y (x + y) d x \Rightarrow \frac{d y}{d x} = - \frac{y}{x} - (\frac{y}{x})^{2}$ , a homogeneous equation.
½ mark
Put $y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$ : $v + x \frac{d v}{d x} = - v - v^{2} \Rightarrow x \frac{d v}{d x} = - (v^{2} + 2 v)$ .
½ mark
Separating variables: $\int \frac{d v}{v ^{2} + 2 v} = - \int \frac{d x}{x}$ .
½ mark
$\int \frac{d v}{( v + 1 ) ^{2} - 1} = - \int \frac{d x}{x}$ .
½ mark
$\frac{1}{2} lo g \frac{v}{v + 2} = - lo g ∣ x ∣ + lo g ∣ C ∣ = lo g \frac{C}{x}$ .
½ mark
So $\frac{v}{v + 2} = \frac{C ^{2}}{x ^{2}}$ ; with $v = \frac{y}{x}$ : $\frac{y}{y + 2 x} = \frac{k}{x ^{2}}$ , i.e. $x^{2} y = k (y + 2 x)$ .
½ mark

Answer: $x^{2} y = k (y + 2 x)$ , k an arbitrary constant

Where the marks are

Writes $\frac{d y}{d x} = - \frac{y}{x} - \frac{y ^{2}}{x ^{2}}$ (½ mark)
Substitutes $y = v x$ (½ mark)
Separates variables (½ mark)
Writes $v^{2} + 2 v = (v + 1)^{2} - 1$ (½ mark)
Integrates correctly (½ mark)
Final solution in x and y (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 paper2024 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 29(b)

Hints

Step-by-step solution

Revise the chapter

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