Mathematics Minor: Differential Equations (MTMMIN303T)

An introductory lecture covering basic definitions and types of first-order ODEs.

A series of videos demonstrating the integrating factor method for linear equations.

Understanding real-world applications of first-order differential equations.

1. The general solution of a first-order differential equation contains:
   A) No arbitrary constant    B) One arbitrary constant    C) Two arbitrary constants    D) Infinite arbitrary constants
   Answer: B
2. Which of the following is a linear first-order differential equation?
   A) $$ yfrac{dy}{dx} = x $$    B) $$ frac{dy}{dx} + y^2 = sin x $$    C) $$ frac{dy}{dx} + xy = e^x $$    D) $$ frac{dy}{dx} = cos y $$
   Answer: C

1. Solve the differential equation: $$ (1+x^2)frac{dy}{dx} = x(1+y^2) $$
2. Solve the linear differential equation: $$ frac{dy}{dx} + frac{y}{x} = x^2 $$
3. Find the orthogonal trajectory of the family of curves $$ x^2 + y^2 = Cx $$.

Learn how to solve higher-order homogeneous linear differential equations.

Detailed examples of the method of undetermined coefficients.

Understanding and applying the method of variation of parameters.

1. The auxiliary equation for $$ y'' - 4y' + 4y = 0 $$ is:
   A) $$ m^2 - 4m + 4 = 0 $$    B) $$ m^2 + 4m + 4 = 0 $$    C) $$ m - 4 = 0 $$
   Answer: A
2. The Wronskian of $$ e^x $$ and $$ xe^x $$ is:
   A) $$ e^{2x} $$    B) $$ x^2e^x $$    C) $$ x^2e^{2x} $$
   Answer: A

1. Find the general solution of $$ y'' - 3y' + 2y = 0 $$.
2. Solve $$ y'' + y = cos x $$ using the method of undetermined coefficients.
3. Solve $$ y'' + y = sec x $$ using the method of variation of parameters.

A series explaining how to solve systems of linear differential equations using matrix methods.

Visual explanation of the Lotka-Volterra system in population dynamics.

Understanding how differential equations model spring-mass systems.

1. Solve the system of differential equations: $$ begin{pmatrix} x' y' end{pmatrix} = begin{pmatrix} 1 & 2 3 & 2 end{pmatrix} begin{pmatrix} x y end{pmatrix} $$
2. Derive the differential equation for a simple spring-mass system with damping.
3. Explain the concept of "phase plane" in the context of predator-prey models.