- 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).