FAD1014: MATHEMATICS II — Tutorial 7

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

Question 1: Order and Degree of Differential Equations

State the order and degree of the following differential equations:

(a) $5\frac{d^2y}{dx^2} - x\frac{dy}{dx} + (1 - x)y = \sin y$

(b) $5(y'') - y = e^x$

(c) $\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right) + y = 0$

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

State whether the following differential equations are separable ($g(y)dy = f(x)dx$) or non-separable.

(a) $\frac{dy}{dx} + x^2y = x$

(b) $y' - x^2y^2 = x^2$

(c) $\frac{dy}{dx} = -\frac{x}{y-3}$

(d) $\frac{dy}{dx} - 2xy = x^2 - x$

Find the general solution of the following differential equations.

(a) $\frac{dy}{dx} - x\sqrt{1+x^2} = 0$

(b) $e^y\frac{dy}{dx} + \sin x = 0$

(c) $(x + 1)\frac{dy}{dx} = x(y + 3)$

(d) $3y^2\frac{dy}{dx} + 2x = 1$

(e) $x , dy - y , dx = 0$

(f) $3y , dx + (xy + 5x) , dy = 0$

(g) $(y + yx^2)dy + (x + xy^2)dx = 0$

Find the particular solution for each of the following differential equations with the given initial conditions.

(a) $\frac{dy}{dx} = 1 - \frac{2}{y}$ ; $y = 3$ when $x = 3$

(b) $(xy^2 - xy)dx - 2dy = 0$ ; $y = 2$ when $x = 0$

(c) $\cos y , dx + x\sin y , dy = 0$ ; $y(3) = \frac{\pi}{3}$

Prove that $y$ is a solution of the differential equation.

(a) $(x - 2y)\frac{dy}{dx} + 2x + y = 0$ ; $y^2 - x^2 - xy = C$

(b) $y\frac{dy}{dx} = x$ ; $x^2 - y^2 = C$

(Hint: differentiate implicitly)

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