math150L2_part1

# math150L2_part1 - Section I Introduction 1 Preliminaries...

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
E.g. Spring-mass system m = mass, x = displacement, k = spring constant s.t. the force of spring = kx from the Hook’s law.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern