Mathematics · Textbook solutions

Differential Equations

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 127 questions

9.2 Basic Concepts — Order and Degree

15 q

Solved Examples

Worked · 3

Find the order and degree, if defined, of each of the following differential equations:
9.1 Eg.1(i) $\frac{dy}{dx} - \cos x = 0$
9.1 Eg.1(ii) $xy\frac{d^{2}y}{dx^{2}} + x\left(\frac{dy}{dx}\right)^{2} - y\frac{dy}{dx} = 0$
9.1 Eg.1(iii) $y''' + y^{2} + e^{y'} = 0$

Exercise 9.1

Practice · 12

Ex 9.1 Q1 $\frac{d^{4}y}{dx^{4}} + \sin(y''') = 0$
Ex 9.1 Q2 $y' + 5y = 0$
Ex 9.1 Q3 $\left(\frac{ds}{dt}\right)^{4} + 3s\frac{d^{2}s}{dt^{2}} = 0$
Ex 9.1 Q4 $\left(\frac{d^{2}y}{dx^{2}}\right)^{2} + \cos\left(\frac{dy}{dx}\right) = 0$
Ex 9.1 Q5 $\frac{d^{2}y}{dx^{2}} = \cos 3x + \sin 3x$
Ex 9.1 Q6 $(y''')^{2} + (y'')^{3} + (y')^{4} + y^{5} = 0$
Ex 9.1 Q7 $y''' + 2y'' + y' = 0$
Ex 9.1 Q8 $y' + y = e^{x}$
Ex 9.1 Q9 $y'' + (y')^{2} + 2y = 0$
Ex 9.1 Q10 $y'' + 2y' + \sin y = 0$
Ex 9.1 Q11 $\left(\frac{d^{2}y}{dx^{2}}\right)^{3} + \left(\frac{dy}{dx}\right)^{2} + \sin\left(\frac{dy}{dx}\right) + 1 = 0$
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Ex 9.1 Q12 $2x^{2}\frac{d^{2}y}{dx^{2}} - 3\frac{dy}{dx} + y = 0$
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9.3 General and Particular Solutions of a Differential Equation

14 q

Solved Examples

Worked · 2

9.2 Eg.2 $y = e^{-3x}$
9.2 Eg.3 $y = a\cos x + b\sin x$

Exercise 9.2

Practice · 12

Ex 9.2 Q1 $y = e^x + 1$
Ex 9.2 Q2 $y = x^2 + 2x + \mathrm{C}$
Ex 9.2 Q3 $y = \cos x + \mathrm{C}$
Ex 9.2 Q4 $y = \sqrt{1 + x^2}$
Ex 9.2 Q5 $y = \mathrm{A}x$
Ex 9.2 Q6 $y = x\sin x$
Ex 9.2 Q7 $xy = \log y + \mathrm{C}$
Ex 9.2 Q8 $y - \cos y = x$
Ex 9.2 Q9 $x + y = \tan^{-1} y$
Ex 9.2 Q10 $y = \sqrt{a^2 - x^2}$
Ex 9.2 Q11 $The number of arbitrary constants in the general solution of a differential equation of fourth order are:$
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Ex 9.2 Q12 $The number of arbitrary constants in the particular solution of a differential equation of third order are:$
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9.4.1 Differential Equations with Variables Separable

29 q

Solved Examples

Worked · 6

9.3 Eg.4 $\frac{dy}{dx} = \frac{x+1}{2-y}$
9.3 Eg.5 $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$
9.3 Eg.6 $\frac{dy}{dx} = -4xy^2$
9.3 Eg.7 $(1, 1)$
9.3 Eg.8 $(-2, 3)$
9.3 Eg.9 $In a bank, principal increases continuously at the rate of 5% per year. In how many years Rs 1000 double itself?$

Exercise 9.3

Practice · 23

Ex 9.3 Q1 $\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}$
Ex 9.3 Q2 $\frac{dy}{dx} = \sqrt{4 - y^2}\ (-2 < y < 2)$
Ex 9.3 Q3 $\frac{dy}{dx} + y = 1\ (y \neq 1)$
Ex 9.3 Q4 $\sec^2 x \tan y\,dx + \sec^2 y \tan x\,dy = 0$
Ex 9.3 Q5 $(e^x + e^{-x})\,dy - (e^x - e^{-x})\,dx = 0$
Ex 9.3 Q6 $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$
Ex 9.3 Q7 $y \log y\,dx - x\,dy = 0$
Ex 9.3 Q8 $x^5 \frac{dy}{dx} = -y^5$
Ex 9.3 Q9 $\frac{dy}{dx} = \sin^{-1} x$
Ex 9.3 Q10 $e^x \tan y\,dx + (1 - e^x)\sec^2 y\,dy = 0$
Ex 9.3 Q11 $(x^3 + x^2 + x + 1)\frac{dy}{dx} = 2x^2 + x$
Ex 9.3 Q12 $x(x^2 - 1)\frac{dy}{dx} = 1$
Ex 9.3 Q13 $\cos\left(\frac{dy}{dx}\right) = a\ (a \in \mathbf{R})$
Ex 9.3 Q14 $\frac{dy}{dx} = y \tan x$
Ex 9.3 Q15 $(0, 0)$
Ex 9.3 Q16 $xy\frac{dy}{dx} = (x + 2)(y + 2)$
Ex 9.3 Q17 $(0, -2)$
Ex 9.3 Q18 $(x, y)$
Ex 9.3 Q19 $t$
Ex 9.3 Q20 $r\%$
Ex 9.3 Q21 $(e^{0.5} = 1.648)$
Ex 9.3 Q22 $In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?$
Ex 9.3 Q23 $\frac{dy}{dx} = e^{x+y}$
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9.4.2 Homogeneous Differential Equations

21 q

Solved Examples

Worked · 4

9.4 Eg.10 $(x - y)\frac{dy}{dx} = x + 2y$
9.4 Eg.11 $x\cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y\cos\left(\frac{y}{x}\right) + x$
9.4 Eg.12 $2y\,e^{\frac{x}{y}}dx + \left(y - 2x\,e^{\frac{x}{y}}\right)dy = 0$
9.4 Eg.13 $(x, y)$

Exercise 9.4

Practice · 17

Ex 9.4 Q1 $(x^{2} + xy)\,dy = (x^{2} + y^{2})\,dx$
Ex 9.4 Q2 $y' = \frac{x + y}{x}$
Ex 9.4 Q3 $(x - y)\,dy - (x + y)\,dx = 0$
Ex 9.4 Q4 $(x^{2} - y^{2})\,dx + 2xy\,dy = 0$
Ex 9.4 Q5 $x^{2}\frac{dy}{dx} = x^{2} - 2y^{2} + xy$
Ex 9.4 Q6 $x\,dy - y\,dx = \sqrt{x^{2} + y^{2}}\,dx$
Ex 9.4 Q7 $\left\{x\cos\left(\frac{y}{x}\right) + y\sin\left(\frac{y}{x}\right)\right\}y\,dx = \left\{y\sin\left(\frac{y}{x}\right) - x\cos\left(\frac{y}{x}\right)\right\}x\,dy$
Ex 9.4 Q8 $x\frac{dy}{dx} - y + x\sin\left(\frac{y}{x}\right) = 0$
Ex 9.4 Q9 $y\,dx + x\log\left(\frac{y}{x}\right)dy - 2x\,dy = 0$
Ex 9.4 Q10 $\left(1 + e^{\frac{x}{y}}\right)dx + e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)dy = 0$
Ex 9.4 Q11 $(x + y)\,dy + (x - y)\,dx = 0$
Ex 9.4 Q12 $x^{2}\,dy + (xy + y^{2})\,dx = 0$
Ex 9.4 Q13 $\left[x\sin^{2}\left(\frac{y}{x}\right) - y\right]dx + x\,dy = 0$
Ex 9.4 Q14 $\frac{dy}{dx} - \frac{y}{x} + \text{cosec}\left(\frac{y}{x}\right) = 0$
Ex 9.4 Q15 $2xy + y^{2} - 2x^{2}\frac{dy}{dx} = 0$
Ex 9.4 Q16 $\frac{dx}{dy} = h\left(\frac{x}{y}\right)$
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Ex 9.4 Q17 $Which of the following is a homogeneous differential equation?$
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9.4.3 Linear Differential Equations

24 q

Solved Examples

Worked · 5

9.5 Eg.14 $\frac{dy}{dx} - y = \cos x$
9.5 Eg.15 $x\frac{dy}{dx} + 2y = x^2$
9.5 Eg.16 $y\,dx - (x + 2y^2)\,dy = 0$
9.5 Eg.17 $\frac{dy}{dx} + y\cot x = 2x + x^2\cot x$
9.5 Eg.18 $(0, 1)$

Exercise 9.5

Practice · 19

Ex 9.5 Q1 $\frac{dy}{dx} + 2y = \sin x$
Ex 9.5 Q2 $\frac{dy}{dx} + 3y = e^{-2x}$
Ex 9.5 Q3 $\frac{dy}{dx} + \frac{y}{x} = x^2$
Ex 9.5 Q4 $\frac{dy}{dx} + (\sec x)\,y = \tan x$
Ex 9.5 Q5 $\cos^2 x\,\frac{dy}{dx} + y = \tan x$
Ex 9.5 Q6 $x\frac{dy}{dx} + 2y = x^2\log x$
Ex 9.5 Q7 $x\log x\,\frac{dy}{dx} + y = \frac{2}{x}\log x$
Ex 9.5 Q8 $(1 + x^2)\,dy + 2xy\,dx = \cot x\,dx$
Ex 9.5 Q9 $x\frac{dy}{dx} + y - x + xy\cot x = 0$
Ex 9.5 Q10 $(x + y)\frac{dy}{dx} = 1$
Ex 9.5 Q11 $y\,dx + (x - y^2)\,dy = 0$
Ex 9.5 Q12 $(x + 3y^2)\frac{dy}{dx} = y$
Ex 9.5 Q13 $\frac{dy}{dx} + 2y\tan x = \sin x$
Ex 9.5 Q14 $(1 + x^2)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}$
Ex 9.5 Q15 $\frac{dy}{dx} - 3y\cot x = \sin 2x$
Ex 9.5 Q16 $(x, y)$
Ex 9.5 Q17 $(0, 2)$
Ex 9.5 Q18 $x\frac{dy}{dx} - y = 2x^2$
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Ex 9.5 Q19 $(1 - y^2)\frac{dx}{dy} + yx = ay$
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Miscellaneous Exercise on Chapter 9

24 q

Solved Examples

Worked · 4

Misc Eg.19 $y = c_1 e^{ax} \cos bx + c_2 e^{ax} \sin bx$
Misc Eg.20 $\log\left(\frac{dy}{dx}\right) = 3x + 4y$
Misc Eg.21 $(x\,dy - y\,dx)\,y\sin\left(\frac{y}{x}\right) = (y\,dx + x\,dy)\,x\cos\left(\frac{y}{x}\right)$
Misc Eg.22 $(\tan^{-1}y - x)\,dy = (1 + y^2)\,dx$

Miscellaneous Exercise

Practice · 20

For each of the differential equations given below, indicate its order and degree (if defined).
Misc Q1(i) $\frac{d^2y}{dx^2} + 5x\left(\frac{dy}{dx}\right)^2 - 6y = \log x$
Misc Q1(ii) $\left(\frac{dy}{dx}\right)^3 - 4\left(\frac{dy}{dx}\right)^2 + 7y = \sin x$
Misc Q1(iii) $\frac{d^4y}{dx^4} - \sin\left(\frac{d^3y}{dx^3}\right) = 0$
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
Misc Q2(i) $xy = a\,e^{x} + b\,e^{-x} + x^2$
Misc Q2(ii) $y = e^{x}(a\cos x + b\sin x)$
Misc Q2(iii) $y = x\sin 3x$
Misc Q2(iv) $x^2 = 2y^2\log y$
Misc Q3 $x^2 - y^2 = c\,(x^2 + y^2)^2$
Misc Q4 $\frac{dy}{dx} + \sqrt{\frac{1 - y^2}{1 - x^2}} = 0$
Misc Q5 $\frac{dy}{dx} + \frac{y^2 + y + 1}{x^2 + x + 1} = 0$
Misc Q6 $\left(0, \frac{\pi}{4}\right)$
Misc Q7 $(1 + e^{2x})\,dy + (1 + y^2)\,e^{x}\,dx = 0$
Misc Q8 $y\,e^{\frac{x}{y}}\,dx = \left(x\,e^{\frac{x}{y}} + y^2\right)dy\;(y \ne 0)$
Misc Q9 $(x - y)\,(dx + dy) = dx - dy$
Misc Q10 $\left[\frac{e^{-2\sqrt{x}}}{\sqrt{x}} - \frac{y}{\sqrt{x}}\right]\frac{dx}{dy} = 1\;(x \ne 0)$
Misc Q11 $\frac{dy}{dx} + y\cot x = 4x\operatorname{cosec} x\;(x \ne 0)$
Misc Q12 $(x + 1)\frac{dy}{dx} = 2e^{-y} - 1$
Misc Q13 $\frac{y\,dx - x\,dy}{y} = 0$
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Misc Q14 $\frac{dx}{dy} + \mathrm{P}_1 x = \mathrm{Q}_1$
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Misc Q15 $e^{x}\,dy + (y\,e^{x} + 2x)\,dx = 0$
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What the board has asked from this chapter

Questions per paper · 26 papers · 2022 to 2026

Bars average the full papers of each year. 53 further questions come from set variants and are listed below.

2026

37 q

4 $x\,dy - y\,dx = 0$
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Variables Separable
5 $\frac{d}{dx} (\sin y) = y^2$
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Order and Degree of a Differential Equation
11 $\dfrac{dy}{dx} = \dfrac{\sqrt{y}}{\sqrt{x}}$
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Variables Separable
11 $R \frac{dx}{dy} + Px = Q$
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Linear Differential Equations
12 $y = x\dfrac{dy}{dx} + 3\left[2 - \left(\dfrac{dy}{dx}\right)^2\right]$
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Order and Degree of a Differential Equation
12 $2x\dfrac{dy}{dx} - y = 3$
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Linear Differential Equations
12 $\frac{d}{dx}(e^y) = 0$
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Order and Degree of a Differential Equation
12 $4\left[\dfrac{d^2y}{dx^2}\right]^2 + 3\left[1 + \left(\dfrac{dy}{dx}\right)^2 - y\right] = 0$
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Order and Degree of a Differential Equation
12 $1 + \left(\dfrac{dy}{dx}\right)^3 = \lambda \left(\dfrac{d^3y}{dx^3}\right)^2$
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Order and Degree of a Differential Equation
13 $Which of the following is not a Linear Differential Equation ?$
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Linear Differential Equations
13 $\dfrac{d^2y}{dx^2} = 1 - \left(\dfrac{d^3y}{dx^3}\right)^2$
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Order and Degree of a Differential Equation
13 $y = \left(\dfrac{d^2y}{dx^2}\right)^2 - \lambda\,\dfrac{dy}{dx}$
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Order and Degree of a Differential Equation
13 $\dfrac{dy}{dx} = F(x, y)$
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Homogeneous Differential Equations
13 $1 + \left(\dfrac{d^3y}{dx^3}\right)^3 = \lambda\,\dfrac{d^2y}{dx^2}$
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Order and Degree of a Differential Equation
14 $\dfrac{dy}{dx} = e^{3x - y}$
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Variables Separable
15 $x^2 dy + y^2 dx = 0$
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Variables Separable
16 $\frac{d}{dx}(y')^3 + (y')^3 = 1$
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Order and Degree of a Differential Equation
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
19 $\dfrac{dy}{dx} = e^{x+y}$
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Variables Separable
27 (a) $(x^2 + y^2)\, dy = xy\, dx$
Homogeneous Differential Equations
27 (a) $y^2\,dx + (x^2 - xy + y^2)\,dy = 0$
Homogeneous Differential Equations
28 $y\,dx + (x - y^3)\,dy = 0$
Linear Differential Equations
28 $(x - \sin y)\,dy + \tan y\,dx = 0$
Linear Differential Equations
28 $(x + 2y^3)\,dy = y\,dx$
Linear Differential Equations
27 (b) $\frac{dy}{dx} = y \tan x$
Variables Separable
27 (b) $\dfrac{dy}{dx} - 3y \cot x = \sin 2x$
Linear Differential Equations
29 (a) $x\dfrac{dy}{dx} = y - x \sin^2 \left(\dfrac{y}{x}\right)$
Homogeneous Differential Equations
29 (a) $x^2 \frac{dy}{dx} = x^2 + xy + y^2$
Homogeneous Differential Equations
29 (b) $y \log y \dfrac{dx}{dy} + x = \dfrac{2}{y}$
Linear Differential Equations
29 (b) $xy \frac{dy}{dx} = (x + 2) \, (y + 2)$
Variables Separable
29 (a) $2x^2 \dfrac{dy}{dx} = y^2 + 2xy$
Homogeneous Differential Equations
30 (a) $(x^2 - y^2)\, dx + 2xy\, dy = 0$
Homogeneous Differential Equations
29 (b) $(x + 1) \dfrac{dy}{dx} = 2\,e^{-y} - 1$
Variables Separable
30 (b) $\sin x \cos y\, dx + \cos x \sin y\, dy = 0$
Variables Separable
30 (a) $(y^2 - x^2)dx = 2xy \, dy$
Homogeneous Differential Equations
30 (b) $(1 + e^{2x})dy + (1 + y^2)e^x\,dx = 0$
Variables Separable
34 (a) $y e^y\, dx = \left(y^3 + 2x e^y\right) dy$
Linear Differential Equations
34 (b) $\left(x^3 - 3xy^2\right) dx = \left(y^3 - 3x^2 y\right) dy$
Homogeneous Differential Equations

2025

32 q

4 $x$
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Homogeneous Differential Equations
14 $\dfrac{dx}{dy} = \dfrac{x \log x}{\dfrac{2}{x} \log x - y}$
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Linear Differential Equations
14 $\dfrac{dx}{dy} = \dfrac{-(1 + \sin x)}{x + y \cos x}$
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Linear Differential Equations
28 $\dfrac{dy}{dx} - \dfrac{y}{x} + \operatorname{cosec}\left(\dfrac{y}{x}\right) = 0$
Homogeneous Differential Equations
28 $x \cos\left(\dfrac{y}{x}\right) \dfrac{dy}{dx} = y \cos\left(\dfrac{y}{x}\right) + x$
Homogeneous Differential Equations
12 $(x + 2y^3)\frac{dy}{dx} = 2y$
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Linear Differential Equations
12 $-\dfrac{d^4y}{dx^4} + 2e^{dy/dx} + y^2 = 0$
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Order and Degree of a Differential Equation
12 $\frac{dy}{dx} + y = \frac{1 + y}{x}$
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Linear Differential Equations
12 $\dfrac{d^2y}{dx^2} + 4\left(\dfrac{dy}{dx}\right) = x \log\left(\dfrac{d^2y}{dx^2}\right)$
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Order and Degree of a Differential Equation
12 $\left[\dfrac{d^2y}{dx^2} - 1\right]^2 = \dfrac{dy}{dx}$
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Order and Degree of a Differential Equation
13 $\log\left(\dfrac{dy}{dx}\right) = 3x + 4y$
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Variables Separable
13 $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^5 = \frac{d^2y}{dx^2}$
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Order and Degree of a Differential Equation
14 $\left(\dfrac{e^{-2\sqrt{x}}}{\sqrt{x}} - \dfrac{y}{\sqrt{x}}\right)\dfrac{dx}{dy} = 1$
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Linear Differential Equations
15 $\left[1 + \left(\dfrac{dy}{dx}\right)^2\right]^3 = \dfrac{d^2y}{dx^2}$
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Order and Degree of a Differential Equation
16 $\frac{dy}{dx} + y \tan x - \sec x = 0$
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Linear Differential Equations
16 $\frac{dy}{dx} = \frac{-x}{y}$
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Variables Separable
16 $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x \sin\left(\frac{dy}{dx}\right)$
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Order and Degree of a Differential Equation
18 $\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)^3 = 0$
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Order and Degree of a Differential Equation
26 (a) $2(y + 3) - xy \dfrac{dy}{dx} = 0$
Variables Separable
26 (b) $(1 + x^2)\dfrac{dy}{dx} + 2xy = 4x^2$
Linear Differential Equations
28 $\left[x\sin^2\left(\dfrac{y}{x}\right) - y\right]dx + x\,dy = 0$
Homogeneous Differential Equations
34 (a) $x^2 y\, dx - (x^3 + y^3)\, dy = 0$
Homogeneous Differential Equations
34 $\dfrac{dy}{dx} = \cos x - 2y$
Linear Differential Equations
34 $(x - \sin y)\, dy + (\tan y)\, dx = 0$
Linear Differential Equations
34 (b) $(1 + x^2)\dfrac{dy}{dx} + 2xy - 4x^2 = 0$
Linear Differential Equations
34 $(x^2 + y^2)\, dx + xy\, dy = 0$
Homogeneous Differential Equations
Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. (Cylindrical-shaped Camphor tablets) A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, $\dfrac{dV}{dt} = kS$ is the differential equation, where V is the volume, S is the surface area and t is the time in hours. Based upon the above information, answer the following questions :
37 (i) $Write the order and degree of the given differential equation.$
Order and Degree of a Differential Equation
Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. (Cylindrical-shaped Camphor tablets) A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, $\dfrac{dV}{dt} = kS$ is the differential equation, where V is the volume, S is the surface area and t is the time in hours. Based upon the above information, answer the following questions :
37 (ii) $V = \pi r^3$
General and Particular Solutions
Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. (Cylindrical-shaped Camphor tablets) A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, $\dfrac{dV}{dt} = kS$ is the differential equation, where V is the volume, S is the surface area and t is the time in hours. Based upon the above information, answer the following questions :
37 (iii) (a) $r = 3$
General and Particular Solutions
Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. (Cylindrical-shaped Camphor tablets) A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, $\dfrac{dV}{dt} = kS$ is the differential equation, where V is the volume, S is the surface area and t is the time in hours. Based upon the above information, answer the following questions :
37 (iii) (b) $r = 1$
General and Particular Solutions
During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature $(25^\circ\text{C})$ . Initially the processor's temperature is $85^\circ\text{C}$ . The rate of cooling is defined by the equation $\frac{d}{dt}(T(t)) = -k(T(t) - 25)$ , where T(t) represents the temperature of the processor at time t (in minutes) and k is a constant. Based on the above information, answer the following questions :
38 (i) $T(0) = 85^\circ\text{C}$
Variables Separable
During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature $(25^\circ\text{C})$ . Initially the processor's temperature is $85^\circ\text{C}$ . The rate of cooling is defined by the equation $\frac{d}{dt}(T(t)) = -k(T(t) - 25)$ , where T(t) represents the temperature of the processor at time t (in minutes) and k is a constant. Based on the above information, answer the following questions :
38 (ii) $40^\circ\text{C}$
Variables Separable

2024

27 q

1 $\dfrac{dy}{dx} - y = 1$
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General and Particular Solutions
2 $(y'')^2 + \log (y') = x^5$
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Order and Degree of a Differential Equation
2 $\frac{dy}{dx} = \frac{1}{\log y}$
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Variables Separable
2 $x \frac{dy}{dx} - y = x^4 - 3x$
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Linear Differential Equations
7 $x \log x \dfrac{dy}{dx} + y = 2 \log x$
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Linear Differential Equations
7 $\dfrac{dy}{dx} = F(x, y)$
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Homogeneous Differential Equations
8 $(1 - x^2) \frac{dy}{dx} + xy = ax$
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Linear Differential Equations
9 $\dfrac{dy}{dx} + \dfrac{2}{x}y = 0$
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Linear Differential Equations
9 $\log\left(\dfrac{dy}{dx}\right) = 3x + 4y;\ y(0) = 0$
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General and Particular Solutions
9 $x\,dy + y\,dx = 0$
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Variables Separable
10 $(x + 2y^2)\dfrac{dy}{dx} = y\ (y > 0)$
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Linear Differential Equations
14 $\dfrac{d^4y}{dx^4} - \sin\left(\dfrac{d^2y}{dx^2}\right) = 5$
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Order and Degree of a Differential Equation
14 $\dfrac{dy}{dx} = e^{x + y}$
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Variables Separable
14 $\left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \frac{d^2y}{dx^2}$
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Order and Degree of a Differential Equation
14 $(y'')^2 + (y')^3 = x \sin (y')$
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Order and Degree of a Differential Equation
14 $\dfrac{\mathrm{d}^3\mathrm{y}}{\mathrm{d}x^3} + x\left(\dfrac{\mathrm{d}\mathrm{y}}{\mathrm{d}x}\right)^5 = 4\log\left(\dfrac{\mathrm{d}^4\mathrm{y}}{\mathrm{d}x^4}\right)$
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Order and Degree of a Differential Equation
27 $\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}$
Homogeneous Differential Equations
28 (a) $2xy + y^2 - 2x^2 \dfrac{dy}{dx} = 0$
Homogeneous Differential Equations
29 (a) $\dfrac{dy}{dx} = y \cot 2x$
Variables Separable
29 (a) $\dfrac{\mathrm{d}y}{\mathrm{d}x} - 2x\mathrm{y} = 3x^2\, \mathrm{e}^{x^2}$
Linear Differential Equations
29 (b) $\left(xe^{y/x} + y\right) dx = x\,dy$
Homogeneous Differential Equations
28 (b) $y\, dx = (x + 2y^2)\, dy$
Linear Differential Equations
29 $x^2 \frac{dy}{dx} - xy = x^2 \cos^2 \left( \frac{y}{2x} \right)$
Homogeneous Differential Equations
29 (b) $x^2\, \mathrm{d}y + \mathrm{y}(x + \mathrm{y})\, \mathrm{d}x = 0$
Homogeneous Differential Equations
31 $(\tan^{-1} y - x) \; dy = (1 + y^2) \; dx$
Linear Differential Equations
A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using exponential growth model, the rate of growth of this sample of bacteria is calculated. The differential equation representing the growth of bacteria is given as : $\dfrac{dP}{dt} = kP$ , where P is the population of bacteria at any time 't'. Based on the above information, answer the following questions :
37 (i) $Obtain the general solution of the given differential equation and express it as an exponential function of 't'.$
Variables Separable
A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using exponential growth model, the rate of growth of this sample of bacteria is calculated. The differential equation representing the growth of bacteria is given as : $\dfrac{dP}{dt} = kP$ , where P is the population of bacteria at any time 't'. Based on the above information, answer the following questions :
37 (ii) $t = 0$
General and Particular Solutions

2023

25 q

6 $\sin x + \cos\left(\frac{dy}{dx}\right) = y^2$
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Order and Degree of a Differential Equation
7 $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 = \sin y$
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Order and Degree of a Differential Equation
7 $(1 - y^2)\frac{dx}{dy} + yx = ay$
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Linear Differential Equations
8 $\frac{dx}{x} + \frac{dy}{y} = 0$
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Variables Separable
9 $\frac{d^2 y}{dx^2} \sin y + \left( \frac{dy}{dx} \right)^3 \cos y = \sqrt{y}$
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Order and Degree of a Differential Equation
10 $x\dfrac{dy}{dx} - y = 2x^2$
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Linear Differential Equations
11 $\left(\dfrac{d^2y}{dx^2}\right)^2 + \left(\dfrac{dy}{dx}\right)^3 = x \sin\left(\dfrac{dy}{dx}\right)$
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Order and Degree of a Differential Equation
12 $\dfrac{d}{dx}\left(\left(\dfrac{dy}{dx}\right)^3\right)$
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Order and Degree of a Differential Equation
12 $\left(1 + 3\dfrac{dy}{dx}\right)^2 = 4\dfrac{d^3y}{dx^3}$
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Order and Degree of a Differential Equation
13 $\dfrac{dy}{dx} = \dfrac{y + 1}{x - 1}$
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General and Particular Solutions
15 $x \, dy - (1 + x^2) \, dx = dx$
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Variables Separable
26 (a) $\frac{d}{dx} (x y^2) = 2y (1 + x^2)$
Linear Differential Equations
26 (a) $\frac{dx}{dy} = \frac{e^{x/y}\left(\frac{x}{y} - 1\right)}{1 + e^{x/y}}$
Homogeneous Differential Equations
26 (b) $\frac{dy}{dx} + \cot x \cdot y = \cos^2 x$
Linear Differential Equations
26 (b) $x e^{\frac{y}{x}} - y + x \frac{dy}{dx} = 0$
Homogeneous Differential Equations
28 (a) $\frac{dy}{dx} + \sec^2 x \cdot y = \tan x \cdot \sec^2 x$
Linear Differential Equations
28 (b) $x \, dy - y \, dx - \sqrt{x^2 + y^2} \, dx = 0$
Homogeneous Differential Equations
29 (a) $\frac{dy}{dx} - \frac{2y}{x} = \sin \frac{1}{x}$
Linear Differential Equations
29 (a) $(xy - x^2) \, dy = y^2 \, dx$
Homogeneous Differential Equations
29 (b) $\frac{dy}{dx} = \sin(x + y) + \sin(x - y)$
Variables Separable
29 (b) $(x^2 + 1)\frac{dy}{dx} + 2xy = \sqrt{x^2 + 4}$
Linear Differential Equations
29 (a) $\dfrac{dy}{dx} = \dfrac{x + y}{x}$
Homogeneous Differential Equations
29 (b) $e^x \tan y\, dx + (1 - e^x)\sec^2 y\, dy = 0$
Variables Separable
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $\dfrac{dy}{dx} = F(x, y)$ is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogenous function of degree n if $F(\lambda x, \lambda y) = \lambda^n F(x, y)$ . To solve a homogeneous differential equation of the type $\dfrac{dy}{dx} = F(x, y) = g\left(\dfrac{y}{x}\right)$ , we make the substitution $y = vx$ and then separate the variables. Based on the above, answer the following questions :
38 (i) $(x^2 - y^2)\,dx + 2xy\,dy = 0$
Homogeneous Differential Equations
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $\dfrac{dy}{dx} = F(x, y)$ is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogenous function of degree n if $F(\lambda x, \lambda y) = \lambda^n F(x, y)$ . To solve a homogeneous differential equation of the type $\dfrac{dy}{dx} = F(x, y) = g\left(\dfrac{y}{x}\right)$ , we make the substitution $y = vx$ and then separate the variables. Based on the above, answer the following questions :
38 (ii) $Solve the above equation to find its general solution.$
Homogeneous Differential Equations

2022

20 q

1 $\left(x + \frac{dy}{dx}\right)^2 = \left(\frac{dy}{dx}\right)^2 + 1$
Order and Degree of a Differential Equation
1 $\left[ \frac{d}{dx} (xy^2) \right] \cdot \frac{dy}{dx} + y = 0$
Order and Degree of a Differential Equation
2 $e^{dy/dx} = x^2$
Variables Separable
2 $\dfrac{dy}{dx} = e^{x-y} + x^2 e^{-y}$
Variables Separable
2 $\log \left( \dfrac{dy}{dx} \right) = ax + by$
Variables Separable
3 $\dfrac{dy}{dx} = \dfrac{3e^{2x} + 3e^{4x}}{e^x + e^{-x}}$
Variables Separable
5 $\sec^2 x \cdot \tan y \, dx + \sec^2 y \cdot \tan x \, dy = 0.$
Variables Separable
5 $\log\left(\frac{dy}{dx}\right) = x - y$
Variables Separable
6 $\frac{dy}{dx} = 2y^2$
Variables Separable
8 (a) $x \dfrac{dy}{dx} - y = x^2 \cdot e^x$
Linear Differential Equations
8 (b) $x \dfrac{dy}{dx} = y(\log y - \log x + 1)$
Homogeneous Differential Equations
8 $x \frac{dy}{dx} - y + x \sin \frac{y}{x} = 0$
Homogeneous Differential Equations
9 $\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$
Homogeneous Differential Equations
9 $(y + 3x^2) \frac{dx}{dy} = x$
Linear Differential Equations
9 $(1 + e^{y/x}) \, dy + e^{y/x} \left( 1 - \frac{y}{x} \right) dx = 0$
Homogeneous Differential Equations
10 $x\frac{dy}{dx} + x\cos^2\left(\frac{y}{x}\right) = y$
Homogeneous Differential Equations
12 (a) $x \dfrac{dy}{dx} + y + \dfrac{1}{1 + x^2} = 0$
Linear Differential Equations
12 (b) $x (y^3 + x^3) \, dy = (2y^4 + 5x^3 y) \, dx.$
Homogeneous Differential Equations
13 (a) $\left(y - \sin^2 x\right) dx + \tan x\, dy = 0$
Linear Differential Equations
13 (b) $\left(x^3 + y^3\right) dy = x^2 y\, dx$
Homogeneous Differential Equations

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 3 times: 2023, 2024, 2026* (* = numbers or wording changed that year)
4 $x\,dy - y\,dx = 0$
Tap an option to check your answer.
Question 2
Asked 3 times: 2023, 2024, 2025* (* = numbers or wording changed that year)
16 $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x \sin\left(\frac{dy}{dx}\right)$
Tap an option to check your answer.
Question 3
Asked 2 times: 2023, 2026
30 (a) $(x^2 - y^2)\, dx + 2xy\, dy = 0$
Question 4
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
29 (a) $x\dfrac{dy}{dx} = y - x \sin^2 \left(\dfrac{y}{x}\right)$
Question 5
Asked 2 times: 2025, 2026
28 $(x - \sin y)\,dy + \tan y\,dx = 0$
Question 6
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
28 $(x + 2y^3)\,dy = y\,dx$
Question 7
Asked 2 times: 2024, 2026
Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the options given below.
19 $\dfrac{dy}{dx} = e^{x+y}$
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Question 8
Asked 2 times: 2022, 2025
34 (a) $x^2 y\, dx - (x^3 + y^3)\, dy = 0$
Question 9
Asked 2 times: 2023*, 2025 (* = numbers or wording changed that year)
26 (b) $(1 + x^2)\dfrac{dy}{dx} + 2xy = 4x^2$
Question 10
Asked 2 times: 2024, 2025
15 $\left[1 + \left(\dfrac{dy}{dx}\right)^2\right]^3 = \dfrac{d^2y}{dx^2}$
Tap an option to check your answer.

9.2 Basic Concepts — Order and Degree

Solved Examples

Exercise 9.1

9.3 General and Particular Solutions of a Differential Equation

Solved Examples

Exercise 9.2

9.4.1 Differential Equations with Variables Separable

Solved Examples

Exercise 9.3

9.4.2 Homogeneous Differential Equations

Solved Examples

Exercise 9.4

9.4.3 Linear Differential Equations

Solved Examples

Exercise 9.5

Miscellaneous Exercise on Chapter 9

Solved Examples

Miscellaneous Exercise

What the board has asked from this chapter

2026

2025

2024

2023

2022

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10