Exam Questions & Step-by-Step Markschemes (5 Problems)
Question 1 • 2024
6 Marks
Solve the homogeneous differential equation: (x^2 + xy)\,dy = (x^2 + y^2)\,dx given that y = 0 when x = 1 .
Answer: (x - y)^2 = x e^{-y/x}
Question 2 • 2024
6 Marks
Solve the differential equation (x^2 - y^2)dx + 2xy\,dy = 0 , given that y = 1 when x = 1 .
Answer: x^2 + y^2 = 2x
Question 3 • 2023
5 Marks
Find the general solution of the differential equation: (1 + x^2) \frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}
Answer: y = \frac{\arctan x + C}{1 + x^2}
Question 4 • 2023
4 Marks
Find the orthogonal trajectories of the family of parabolas y^2 = 4ax , where a is an arbitrary parameter.
Answer: 2x^2 + y^2 = K \quad (\text{Family of Ellipses})
Question 5 • 2024
6 Marks
Solve the differential equation: (x^2 + 1) \frac{dy}{dx} + 2xy = \sqrt{x^2 + 4} given that y = 0 when x = 0 .
Answer: y = \frac{1}{x^2 + 1}\left[ \frac{x}{2}\sqrt{x^2 + 4} + 2\ln\left(\frac{x + \sqrt{x^2 + 4}}{2}\right) \right]