Solve the differential equation: y+d/dx(xy)=x( x+x)

x²y=-x² x+2x x+2 x+(x^4)/(4)+C

MathsCBSE Class 12 · Sample paper, session 2025-26 · 5 marks · Long answer

CBSE Class 12 Mathematics sample paper 2025-26: question 34(A)

From the official CBSE sample question paper for 2025-26, section D, question 34(A), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(A) · 5 marks · Long answerDifferential Equations

Solve the differential equation: $y + \frac{d}{d x} (x y) = x (sin x + x)$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Expand $\frac{d}{d x} (x y)$ using the product rule and collect the $y$ terms.

Hint 2

Put the equation in the form $\frac{d y}{d x} + P y = Q$ and find $e^{\int P d x}$ .

Hint 3

You will need integration by parts twice for $\int x^{2} sin x d x$ .

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.

Expand the derivative of the product:
$y + (x \frac{d y}{d x} + y) = x (sin x + x) \Rightarrow x \frac{d y}{d x} + 2 y = x (sin x + x)$ .
Divide by $x$ :
$\frac{d y}{d x} + \frac{2}{x} y = sin x + x$ .
1 mark
This is a linear equation $\frac{d y}{d x} + P y = Q$ with $P = \frac{2}{x}$ and $Q = sin x + x$ .
Integrating factor:
$I.F. = e^{\int \frac{2}{x} d x} = e^{2 l o g x} = x^{2}$ .
1 mark
The solution is $y \cdot I.F. = \int Q \cdot I.F. d x$ , so
$y x^{2} = \int (sin x + x) x^{2} d x = \int x^{2} sin x d x + \int x^{3} d x$ .
1 mark
Integrate $\int x^{2} sin x d x$ by parts:
$\int x^{2} sin x d x = - x^{2} cos x + 2 \int x cos x d x$ .
Also $\int x^{3} d x = \frac{x ^{4}}{4}$ .
1 mark
By parts again, $\int x cos x d x = x sin x + cos x$ .
Hence the general solution is
$y x^{2} = - x^{2} cos x + 2 (x sin x + cos x) + \frac{x ^{4}}{4} + C .$
1 mark

Answer: $x^{2} y = - x^{2} cos x + 2 x sin x + 2 cos x + \frac{x ^{4}}{4} + C$

Where the marks are

Reduces the equation to $\frac{d y}{d x} + \frac{2}{x} y = sin x + x$ (1 mark)
Identifies $P$ , $Q$ and finds I.F. $= x^{2}$ (1 mark)
Writes $y x^{2} = \int (sin x + x) x^{2} d x$ (1 mark)
Applies integration by parts once and integrates $x^{3}$ correctly (1 mark)
Completes the integration and states the general solution with constant (1 mark)

Revise the chapter

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

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 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 34(A)

Hints

Step-by-step solution

Revise the chapter

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