Introduction

• Preliminary examples, notations and terminologies.

• Elememtary calculus of variation.

Linear Equations of First Order

• Existence and uniqueness of solution, comparison principle, Bernoulli equations.

• Seperable 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, Gronwall's inequality.

• Elementary difference equation, equilibrium point, criteria for stability.

Linear equations of Second Order

• Equivalent linear system of first order.

• Existence and uniqueness of a global solution.

• 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 paramenters.

• Vibration and equilibrium, free motion and forced motion, resonance.

• Non-linear equations: existence and uniqueness. Examples including pursuit curve, suspension curve, etc.

Higher order linear equations

• General solutions, Wronskian determinant.

• Method of undetermined coefficient and variation of parameters.

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.

• Faltung theorem, convolution.

• Elementary Volterra integral equation, asymptotic analysis.