Ech 140 Lecture 1 Overview

Undergraduate Lectures on ECH140: Mathema9cal Methods for Chemical Engineers

June 8-- 23, 2015

Chapter 1: Lecture 1: Overview

Brian G. Higgins Department of Chemical Engineering and

Materials Science University of California, Davis

Course Outline Lecture Schedule:

Morning Lectures: Monday through Friday: 9:00-12:00 A

Review of Ordinary Dieren9al Equa9ons (ODEs)

ClassicaIon of ODEs Concept of independent and dependent variables

Concept of homogeneous versus non-homogeneous ODEs

First Order ODEs Method of soluIons

IntegraIng factors

Classica9on of ODEs

IdenIfy the dependent variable, i.e., y IdenIfy the independent variable, i.e., t Determine if the equaIon is linear or nonlinear in the

dependent variable Determine if the equaIon has constant coecients or

variable coecients Determine if the equaIon is homogeneous or

inhomogeneous

BG Higgins/UC Davis, Dept. of Chem. Eng. & Mat. Sci./Hanoi-Vietnam/September 2012.

Independent versus Dependent Variables

Which are the dependent and independent variables?

Consider the following funcIon:

Solving for a:

Solving for b:

Solving for x:

First Order ODEs Example of an inhomogeneous ODE

SoluIon can be wri^en as a linear combinaIon of the homogeneous part and plus a par4cular solu4on

Consider the homogeneous part of ODE:

Since this is a linear equaIon with constant coecients, the soluIon is an exponenIal. Thus our guess for the soluIon is

inhomogeneous part

par9cular solu9on homogeneous solu9on

For a non-trivial soluIon we require:

BG Higgins/UC Davis, Dept. of Chem. Eng. & Mat. Sci./Hanoi-Vietnam/June 2015.

SubsItuIng back into the ODE gives

c ex(+ 2) = 0

First Order ODEs cont.


inhomogeneous part

Example of an inhomogeneous ODE



Thus the homogeneous soluIon is:

To determine a par4cular solu4on, we guess the soluIon by inspecIon:

SubsItuIng into the ODE gives:

Factoring out gives:

First Order ODEs cont.

inhomogeneous part

Example of an inhomogeneous ODE



Thus the parIcular soluIon is:

Thus the general soluIon of the ODE is:


Note the constant c needs to be determined from the iniIal condiIon (data) for the ODE.


Method of Integra9ng Factors Consider the following inhomogeneous ODE

To solve this system we seek a funcIon such that

Suppose we can nd such a funcIon such that the above is true, then from the ODE we have:

Then integraIng gives

Solving for gives:

Constant of integra9on


How to Determine the Integra9ng Factor?


The key is to nd a funcIon such that

If the above statement is true, then the RHS is

Thus

Simplifying gives:

IntegraIng gives

Note the constant is immaterial and factors out of the soluIon

Constant of integra9on

Examples

For this example:

Example 1:

Solve:

Find the integraIng factor:

SoluIon:

Evaluate integrals:

homogeneous part inhomogeneous part


Examples Example 2:

Evaluate constant of integraIon:

For this example:

Find the integraIng factor:

Simplify:


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Ech 140 Lecture 1 Overview