Differential Equations: Course Outline#

This outline provides a standard structure for a first university-level course in ordinary differential equations (ODEs).

Part 1: First-Order Differential Equations#

Introduction and Basic Concepts#

1.1 Motivation and History: What is a differential equation? Where do they come from? (Historical context, modeling examples from physics, biology, and engineering).

1.2 Terminology: Definition of a DE, order, linearity, solution (general, particular, singular), initial value problems (IVPs).

1.3 Direction Fields: Visualizing solutions without solving. Existence and uniqueness of solutions.

Methods for Solving First-Order Equations#

2.1 Separable Equations: The simplest class of ODEs.

2.2 Linear Equations: Method of integrating factors.

2.3 Exact Equations: Test for exactness and method of solution.

Substitutions and Further Methods#

3.1 Homogeneous Equations: Substitution methods.

3.2 Bernoulli Equations: Transforming into linear equations.

3.3 Riccati Equations: Advanced solution techniques.

Applications of First-Order ODEs#

4.1 Population Dynamics: Exponential and logistic growth models.

4.2 Mixing Problems: Modeling concentration in tanks.

4.3 Newton's Law of Cooling & Heating.

4.4 Basic Electrical Circuits: $RC$ and $RL$ circuits.

Part 2: Higher-Order Linear Differential Equations#

Introduction to Second-Order Linear Equations#

5.1 Theory of Linear Equations: Principle of superposition, linear independence, the Wronskian.

5.2 Homogeneous Equations with Constant Coefficients: The characteristic equation (real roots, complex roots, repeated roots).

Nonhomogeneous Equations#

6.1 Method of Undetermined Coefficients: Solving for specific forms of the nonhomogeneous term (polynomials, exponentials, sines/cosines).

6.2 Superposition for Nonhomogeneous Equations.

Advanced Methods for Nonhomogeneous Equations#

7.1 Variation of Parameters: A general method for any nonhomogeneous term.

7.2 Higher-Order Linear Equations: Extending the constant coefficient method to order $n > 2$.

Applications of Second-Order Equations#

8.1 Mechanical Vibrations: Mass-spring systems (free undamped, free damped, forced motion).

8.2 Resonance: The phenomenon of forced vibrations at the natural frequency.

8.3 Electrical Circuits: $LRC$ circuits.

Cauchy-Euler Equations#

9.1 The Cauchy-Euler Equation: A special type of variable-coefficient equation.

9.2 Reduction of Order.

Part 3: The Laplace Transform#

Definition and Properties of the Laplace Transform#

10.1 Definition and Existence: The integral definition of the Laplace transform.

10.2 Transforms of Basic Functions.

10.3 Properties: Linearity, first and second shifting theorems.

Solving IVPs with the Laplace Transform#

11.1 Transforms of Derivatives and Integrals.

11.2 The Inverse Laplace Transform: Using partial fraction decomposition.

11.3 Solving IVPs: Transforming the entire differential equation into an algebraic problem.

Step Functions, Impulses, and Convolutions#

12.1 Heaviside (Unit Step) Function: Transforms of piecewise-continuous functions.

12.2 Dirac Delta Function: Modeling impulses.

12.3 Convolution Theorem.

Part 4: Systems of Linear Differential Equations#

Introduction to Systems#

13.1 Systems and Matrix Notation: Converting higher-order ODEs into first-order systems.

13.2 Homogeneous Linear Systems with Constant Coefficients: Eigenvalue method for finding solutions.

Phase Plane Analysis#

14.1 Real Eigenvalues: Node and saddle point portraits.

14.2 Complex and Repeated Eigenvalues: Spiral and center portraits.

14.3 Nonhomogeneous Systems: Introduction to solving nonhomogeneous systems.