Find the particular solution of the differential equation dy/dx-2xy=3x²e^x²; y(0)=5.

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

CBSE Class 12 Mathematics 2024: question 29(a)

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, 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 $\frac{d y}{d x} - 2 x y = 3 x^{2} e^{x^{2}}$ ; $y (0) = 5$ .

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

Compare with $\frac{d y}{d x} + P y = Q$ .

Hint 2

Integrating factor is $e^{\int P d x}$ ; notice how the exponentials cancel in the integral.

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.

The equation is linear of the form $\frac{d y}{d x} + P y = Q$ with $P = - 2 x$ , $Q = 3 x^{2} e^{x^{2}}$ .
½ mark
I.F. $= e^{\int - 2 x d x} = e^{- x^{2}}$ .
1 mark
Solution: $y e^{- x^{2}} = \int e^{- x^{2}} \cdot 3 x^{2} e^{x^{2}} d x + C = \int 3 x^{2} d x + C \Rightarrow y e^{- x^{2}} = x^{3} + C$ .
1 mark
$y (0) = 5$ : $5 \cdot 1 = 0 + C \Rightarrow C = 5$ . Particular solution: $y = (x^{3} + 5) e^{x^{2}}$ .
½ mark

Answer: $y = (x^{3} + 5) e^{x^{2}}$

Where the marks are

Identifies P and Q (½ mark)
Integrating factor $e^{- x^{2}}$ (1 mark)
General solution $y e^{- x^{2}} = x^{3} + C$ (1 mark)
Uses $y (0) = 5$ to get $C = 5$ and 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 paper2024 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 29(a)

Hints

Step-by-step solution

Revise the chapter

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