Solve the differential equation dy/dx = x - 2y.

MathsCBSE Class 12 · Board exam 2025, set 65/7/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2025: question 34

Set in the CBSE Class 12 board exam of 2025, paper 65/7/1, section D, question 34, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34 · 5 marks · Long answerDifferential Equations

Solve the differential equation $\frac{d y}{d x} = cos x - 2 y$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/7/1.

Hints

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

Hint 1

Move the $2 y$ term to the left side; what type of first-order equation do you get?

Hint 2

For $\frac{d y}{d x} + P y = Q$ , the integrating factor is $e^{\int P d x}$ .

Hint 3

The integral $\int e^{2 x} cos x d x$ needs integration by parts twice; the original integral reappears, so solve for it.

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.

Rewrite the equation in linear form: $\frac{d y}{d x} + 2 y = cos x$ . This is of the type $\frac{d y}{d x} + P y = Q$ with $P = 2$ and $Q = cos x$ .
½ mark
Integrating factor: $I.F. = e^{\int 2 d x} = e^{2 x}$ .
1 mark
The solution is $y \cdot e^{2 x} = \int e^{2 x} cos x d x + C$ .
1½ marks
Let $I_{1} = \int e^{2 x} cos x d x$ . Integrate by parts, taking $cos x$ as the first function:
$I_{1} = cos x \cdot \frac{e ^{2 x}}{2} + \frac{1}{2} \int sin x e^{2 x} d x = \frac{e ^{2 x} c o s x}{2} + \frac{1}{2} [sin x \cdot \frac{e ^{2 x}}{2} - \int cos x \cdot \frac{e ^{2 x}}{2} d x]$
$\Rightarrow I_{1} = \frac{e ^{2 x} c o s x}{2} + \frac{e ^{2 x} s i n x}{4} - \frac{1}{4} I_{1}$
$\Rightarrow \frac{5}{4} I_{1} = \frac{e ^{2 x}}{4} (2 cos x + sin x) \Rightarrow I_{1} = \frac{e ^{2 x}}{5} (2 cos x + sin x)$ .
1½ marks
Hence $y e^{2 x} = \frac{e ^{2 x}}{5} (2 cos x + sin x) + C$ , i.e. the general solution is $y = \frac{1}{5} (2 cos x + sin x) + C e^{- 2 x}$ .
½ mark

Answer: $y = \frac{1}{5} (2 cos x + sin x) + C e^{- 2 x}$

Where the marks are

Writes the equation as $\frac{d y}{d x} + 2 y = cos x$ and identifies $P = 2$ , $Q = cos x$ (½ mark)
Finds I.F. $= e^{2 x}$ (1 mark)
Writes the solution form $y e^{2 x} = \int e^{2 x} cos x d x + C$ (1½ marks)
Evaluates $\int e^{2 x} cos x d x = \frac{e ^{2 x}}{5} (2 cos x + sin x)$ by parts (1½ marks)
Writes the general solution $y = \frac{1}{5} (2 cos x + sin x) + C e^{- 2 x}$ (½ 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 paper2025 board paper, set 65/7/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 34

Hints

Step-by-step solution

Revise the chapter

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