Find the particular solution of the differential equation x dy+(y-x²e^x)dx=0, given that y(1)=0

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

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

From the official CBSE sample question paper for 2026-27, 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

Find the particular solution of the differential equation $x d y + (y - x^{2} e^{x}) d x = 0$ , given that $y (1) = 0$

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

Rewrite the equation in the form $\frac{d y}{d x} + P y = Q$ .

Hint 2

Use the integrating factor $e^{\int P d x}$ , then integrate $x^{2} e^{x}$ by parts twice.

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 $x d x$ : $\frac{d y}{d x} + \frac{1}{x} y = x e^{x}$ , a linear differential equation with $P = \frac{1}{x}$ and $Q = x e^{x}$ .
½ mark
I.F. $= e^{\int \frac{1}{x} d x} = e^{l o g x} = x$
½ mark
The solution is $y \cdot x = \int x \cdot x e^{x} d x = \int x^{2} e^{x} d x$ . By parts: $x y = x^{2} e^{x} - \int 2 x e^{x} d x$
½ mark
By parts again: $x y = x^{2} e^{x} - 2 (x e^{x} - e^{x}) + C$
½ mark
Using $y (1) = 0$ : $0 = e - 2 (e - e) + C \Rightarrow C = - e$
½ mark
The particular solution is $x y = e^{x} (x^{2} - 2 x + 2) - e$
½ mark

Answer: $x y = e^{x} (x^{2} - 2 x + 2) - e$

Where the marks are

Writes the equation in linear form (½ mark)
Finds I.F. $= x$ (½ mark)
Sets up $x y = \int x^{2} e^{x} d x$ and applies integration by parts once (½ mark)
Completes the integration (½ mark)
Finds $C = - e$ (½ mark)
States the particular solution (½ mark)

Revise the chapter

Chapter: Differential Equations (notes, formulas, where marks are lost and every past question)
Topic: Linear 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(A)

Hints

Step-by-step solution

Revise the chapter

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