FAD1014: MATHEMATICS II — Tutorial 10

Centre for Foundation Studies in Science, Universiti Malaya
Session 2025/2026

Question 1: Standard Form and Integrating Factor

Write the following differential equation in the form of: $$\frac{dy}{dx} + P(x)y = Q(x) \quad \text{or} \quad \frac{dx}{dy} + P(y)x = Q(y)$$

Hence, find the integrating factor.

(a) $x,dy + (y + xy - e^x),dx = 0$

(b) $dx = -(3 + 2x\tan y),dy$

(c) $(1 - x)y' + y = x - x^2$

(d) $(2xy - 2),dy + (1 + y^2),dx = 0$

Based on the answer in Question 1, solve and find the general solution of the differential equations.

Solve the following linear first order differential equation:

(a) $3\frac{dy}{dx} + 12y = 4$

(b) $\frac{dy}{dx} = x - y$

(c) $y' + 3x^2y = x^2$

Find the particular solution of each of the following differential equation:

(a) $x,dy - y,dx = x^3e^x,dx$ ; $y = 0$ when $x = 1$

(b) $\frac{dy}{dx} + \frac{y}{1+x} = 3$ ; $y(-1) = 0$

(c) $\frac{dy}{dx} + x\cot y = \sin y$ ; $y = 0$ when $x = 3$

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