Chapter 9

Differential Equations

Every example and exercise of the chapter, worked in full. Each page gives the question, the complete solution, the reason behind every step, and the basics it rests on.

Interactive lesson All chapters Past paper questions (31)

Examples

Exercise 9.1

Exercise 9.2

Exercise 9.3

Exercise 9.3 · Q1General solution of $\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}$ worked
Exercise 9.3 · Q2General solution of $\frac{dy}{dx} = \sqrt{4 - y^2}$ ⁠, $-2 < y < 2$ worked
Exercise 9.3 · Q3General solution of $\frac{dy}{dx} + y = 1$ ⁠, $y \ne 1$ worked
Exercise 9.3 · Q4General solution of $\sec^2x\tan y\,dx + \sec^2y\tan x\,dy = 0$ worked
Exercise 9.3 · Q5General solution of $(e^x + e^{-x})\,dy - (e^x - e^{-x})\,dx = 0$ worked
Exercise 9.3 · Q6General solution of $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ worked
Exercise 9.3 · Q7General solution of $y\log y\,dx - x\,dy = 0$ worked
Exercise 9.3 · Q8General solution of $x^5\frac{dy}{dx} = -y^5$ worked
Exercise 9.3 · Q9General solution of $\frac{dy}{dx} = \sin^{-1}x$ worked
Exercise 9.3 · Q10General solution of $e^x\tan y\,dx + (1 - e^x)\sec^2y\,dy = 0$ worked
Exercise 9.3 · Q11Particular solution of $(x^3 + x^2 + x + 1)\frac{dy}{dx} = 2x^2 + x$ ⁠, $y(0) = 1$ worked
Exercise 9.3 · Q12Particular solution of $x(x^2 - 1)\frac{dy}{dx} = 1$ ⁠, $y(2) = 0$ worked
Exercise 9.3 · Q13Particular solution of $\cos\left(\frac{dy}{dx}\right) = a$ ⁠, $y(0) = 1$ worked
Exercise 9.3 · Q14Particular solution of $\frac{dy}{dx} = y\tan x$ ⁠, $y(0) = 1$ worked
Exercise 9.3 · Q15Curve through $(0, 0)$ with $y' = e^x\sin x$ worked
Exercise 9.3 · Q16 $xy\frac{dy}{dx} = (x + 2)(y + 2)$ through $(1, -1)$ worked
Exercise 9.3 · Q17Curve through $(0, -2)$ with $y\frac{dy}{dx} = x$ worked
Exercise 9.3 · Q18Tangent slope twice that of the segment to $(-4, -3)$ ⁠, through $(-2, 1)$ worked
Exercise 9.3 · Q19Balloon with volume changing at a constant rate: $r = (63t + 27)^{1/3}$ worked
Exercise 9.3 · Q20Continuous interest: Rs 100 doubles in 10 years, so $r = 10\log_e 2 \approx 6.93$ worked
Exercise 9.3 · Q21Rs 1000 at 5% continuous growth is $1000e^{0.5}$ after 10 yearsworked
Exercise 9.3 · Q22Bacteria up 10% in 2 hours: 2,00,000 after $\frac{2\log 2}{\log(11/10)}$ hoursworked
Exercise 9.3 · Q23 (MCQ)General solution of $\frac{dy}{dx} = e^{x + y}$ worked

Exercise 9.4

Exercise 9.5

Miscellaneous Exercise

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