Differential Equations I MATB44H3F

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Differential Equations I MATB44H3F cover the following topics of Differential Equations.

1. Introduction
Preliminaries, Sample Application of Differential Equations

2. First Order Ordinary Differential Equations
Separable Equations, Exact Differential Equations, Integrating Factors, Linear First Order Equations, Substitutions (Bernoulli Equation, Homogeneous Equations, Substitution to Reduce Second Order Equations to First), Order

3. Applications and Examples of First Order ode’s
Orthogonal Trajectories, Exponential Growth and Decay, Population Growth, Predator-Prey Models, Newton’s Law of Cooling, Water Tanks, Motion of Objects Falling Under Gravity with Air Resistance, Escape Velocity, Planetary Motion, Particle Moving on a Curve

4. Linear Differential Equations
Homogeneous Linear Equations, Linear Differential Equations with Constant Coefficients, Nonhomogeneous Linear Equations

5. Second Order Linear Equations
Reduction of Order, Undetermined Coefficients, Shortcuts for Undetermined Coefficients, Variation of Parameters

6. Applications of Second Order Differential Equations
Motion of Object Hanging from a Spring, Electrical Circuits

7. Higher Order Linear Differential Equations
Undetermined Coefficients, Variation of Parameters, Substitutions: Euler’s Equation

8. Power Series Solutions to Linear Differential Equations
Introduction, Background Knowledge Concerning Power Series, Analytic Equations, Power Series Solutions: Levels of Success, Level 1: Finding a finite number of coefficients, Level 2: Finding the recursion relation, Solutions Near a Singular Point, Functions Defined via Differential Equations (Chebyshev Equation, Legendre Equation, Airy Equation, Laguerre’s Equation, Bessel Equation)

9. Linear Systems
Preliminaries, Computing e^T, The 2 × 2 Case in Detail, The Non-Homogeneous Case, Phase Portraits, Real Distinct Eigenvalues, Complex Eigenvalues, Repeated Real Roots

10. Existence and Uniqueness Theorems
Picard’s Method, Existence and Uniqueness Theorem for First Order ODE’s, Existence and Uniqueness Theorem for Linear First Order ODE’s, Existence and Uniqueness Theorem for Linear Systems

11. Numerical Approximations
Euler’s Method, Error Bounds, Improved Euler’s Method, Runge-Kutta Methods

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