Solve the differential equation: ye^x/ydx= (xe^x/y+y² )dy, (y≠0).

MathsCBSE Class 12 · Sample paper, session 2023-24 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2023-24: question 29(a)

From the official CBSE sample question paper for 2023-24, section C, question 29(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(a) · 3 marks · Short answerDifferential Equations

Solve the differential equation: $y e^{x / y} d x = (x e^{x / y} + y^{2}) d y$ , $(y \neq = 0)$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

Collect the terms containing $e^{x / y}$ on one side.

Hint 2

$\frac{y d x - x d y}{y ^{2}}$ is the differential of $\frac{x}{y}$ . (You could also put $x = v y$ , since the equation is homogeneous.)

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.

Rearrange: $e^{x / y} (y d x - x d y) = y^{2} d y \Rightarrow e^{x / y} (\frac{y d x - x d y}{y ^{2}}) = d y$ .
1 mark
Since $d (\frac{x}{y}) = \frac{y d x - x d y}{y ^{2}}$ , this becomes $e^{x / y} d (\frac{x}{y}) = d y$ .
1 mark
Integrating both sides: $\int e^{x / y} d (\frac{x}{y}) = \int d y \Rightarrow e^{x / y} = y + C$ , where $C$ is an arbitrary constant.
1 mark

Answer: $e^{x / y} = y + C$

Where the marks are

Rearranges to $e^{x / y} \frac{y d x - x d y}{y ^{2}} = d y$ (1 mark)
Recognises $d (x / y)$ (1 mark)
Integrates to $e^{x / y} = y + C$ (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 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 29(a)

Hints

Step-by-step solution

Revise the chapter

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