math150L2_part1

math150L2_part1 - Section I Introduction 1 Preliminaries...

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Section I - Introduction 1. Preliminaries 1.1 Function of single variable, y = f ( x ) . y is called dependent variable . x is called independent variable . Derivative of y defined as: dy dx = lim x 0 f ( x + x ) - f ( x ) x dy dx is also called ordinary derivative of y for a single-variable function, usually denoted by y ( x ) . 1.2 Function of several variables, y = f ( x 1 , x 2 , · · · , x n ) . y is called dependent variable . x 1 , x 2 , · · · , x n are called independent variables . For a 3-variable function w = f ( x, y, z ) , the partial derivative of w with respect to x is defined as ∂w ∂x = lim x 0 f ( x + x,y,z ) - f ( x,y,z ) x , i.e., rate of change of w due to the change in x , with variables y, z fixed. 1
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2. Definition Ordinary Differential Equations (ODE) are equations contain ordinary derivatives only. Partial Differential Equations (PDE) are equations contain partial derivatives. # We will consider ODE only in this course! 3. Where are ODE come from when studying physical phenomena through mathematical modelling. physical phenomenon mathematical modelling ODE example of a simplest ODE free falling F = ma mg = m dv dt g = v ( t ) v = ´ gdt + c = gt + c # Solving ODE always involves integration! In the more realistic case when air resistance (drag) is considered, the ODE for the free falling problem is more complicated. mathematical modelling of drag force, drag = γv (proportional to velocity), Newton’s law yields mg - γv = m dv dt dv dt = g - γ m v E.g. Pendulum v = tangential velocity, l = length of string, θ = angular displacement v = l dt , a = dv dt = l d 2 θ d 2 t Balance of force ma = - mgsinθ ml d 2 θ d 2 t = - mgsinθ θ + g l sinθ = 0 (non-linear ODE) for small θ , θ + g l θ = 0 (linear ODE) 2
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E.g. Spring-mass system m = mass, x = displacement, k = spring constant s.t. the force of spring = kx from the Hook’s law.
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