Exam Questions & Step-by-Step Markschemes (5 Problems)
Question 1 • 2023
4 Marks
Find the general solution of the differential equation: \frac{dy}{dx} + 2y \tan x = \sin x, \quad \text{given } y = 0 \text{ when } x = \frac{\pi}{3}
Answer: y = \cos x - 2\cos^2 x
Question 2 • 2023
4 Marks
Find the general solution of the differential equation: \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x \quad (x \neq 0)
Answer: y = x^2 + C \csc x
Question 3 • 2023
4 Marks
Find the general solution of the differential equation: x \frac{dy}{dx} + 2y = x^2 \log x, \quad (x > 0)
Answer: y = \frac{x^2}{4}\left(\log x - \frac{1}{4}\right) + \frac{C}{x^2}
Question 4 • 2022
4 Marks
Solve the differential equation (x - y)\,dy - (x + y)\,dx = 0 .
Answer: \arctan\left(\frac{y}{x}\right) = \frac{1}{2}\ln(x^2 + y^2) + C
Question 5 • 2024
4 Marks
Solve the homogeneous differential equation: x \frac{dy}{dx} - y + x \sin\left(\frac{y}{x}\right) = 0 given that y = \frac{\pi}{2} when x = 1 .
Answer: \csc(y/x) - \cot(y/x) = \frac{1}{x}