Find the particular solution of the differential equation dy/dx + ² x · y = x · ² x, given that y(0) = 0.

MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2023: question 28(a)

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, section C, question 28(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(a) · 3 marks · Short answerDifferential Equations

Find the particular solution of the differential equation $\frac{d y}{d x} + sec^{2} x \cdot y = tan x \cdot sec^{2} x$ , given that $y (0) = 0$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

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

Hint 1

Identify $P (x)$ and $Q (x)$ in $\frac{d y}{d x} + P y = Q$ .

Hint 2

After finding the I.F., substitute $t = tan x$ and integrate by parts.

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.

This is a linear equation with $P = sec^{2} x$ and $Q = tan x sec^{2} x$ . So I.F. $= e^{\int s e c^{2} x d x} = e^{t a n x}$ .
½ mark
The solution is $y e^{t a n x} = \int tan x sec^{2} x e^{t a n x} d x + C$ .
½ mark
Put $tan x = t$ , so $sec^{2} x d x = d t$ .
½ mark
Integrating by parts: $\int t e^{t} d t = t e^{t} - e^{t} = e^{t} (t - 1)$ .
½ mark
Hence $y e^{t a n x} = e^{t a n x} (tan x - 1) + C$ .
½ mark
Using $y (0) = 0$ : $0 = 1 \cdot (0 - 1) + C$ , so $C = 1$ .
The particular solution is $y e^{t a n x} = e^{t a n x} (tan x - 1) + 1$ , i.e. $y = tan x - 1 + e^{- t a n x}$ .
½ mark

Answer: $y = tan x - 1 + e^{- t a n x}$

Where the marks are

I.F. $= e^{t a n x}$ (½ mark)
Writes the solution form $y \cdot$ I.F. $= \int Q \cdot$ I.F. $d x$ (½ mark)
Substitution $tan x = t$ (½ mark)
$\int t e^{t} d t = e^{t} (t - 1)$ (½ mark)
General solution (½ mark)
Finds $C = 1$ and writes 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 paper2023 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 28(a)

Hints

Step-by-step solution

Revise the chapter

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