Course Outline:

• Introduction
• Preliminary examples, notation and terminology.

• Linear Equations of First Order
• Existence and uniqueness of solutions, comparison principle, Bernoulli equation.
• Separable equations, homogeneous equations.
• Exact equations, sufficient and necessary condition for exactness, integrating factor -- making an exact equation.
• Autonomous equations, equilibrium solution, stability, criteria for stability.
• Non-linear equations of first order, fundamental theorem of existence and uniqueness of local solution.
• Elementary difference equations, equilibrium point, criteria for stability.
• Numerical approximation and Euler's method.

• Linear Equations of Second Order
• Equivalent linear system of first order.
• Principle of superposition, Wronskian determinant, fundamental set of solutions, linear independence. Equivalent conditions for fundamental set of solutions.
• General solutions for homogeneous equations with constant coefficients, and their geometric properties. Method of reduction of order.
• General solutions for non-homogeneous equations with constant coefficients. Method of undetermined coefficients and variation of parameters.
• Vibration and equilibrium, free motion and forced motion, resonance.
• Non-linear equations: existence and uniqueness.
• Application of differential equations: elementary calculus of variation.
• Application of differential equations: examples including competitive species, Newton's inverse square law, etc.

• Laplace Transform
• Improper integrals, definition of Laplace transform. Some basic theorems.
• Inverse Laplace transform, one-to-one correspondence between function and its Laplace transform.
• Partial fraction expansion, solving initial value problem by Laplace transform method.
• Differential equations with discontinuous forcing functions.
• Faltung theorem, convolution.
• Elementary Volterra integral equation, asymptotic analysis.

• Introduction to Probability and Statistics (there will be handouts).