Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§9.1 ODE

Models



Review §

Any QUESTIONS About• §8.3 → TrigonoMetric Applications

Any QUESTIONS About HomeWork• §8.3 → HW-12

8.3

“TriAnguLation

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§9.1 Learning Goals Solve “variable separable” differential

equations and initial value problems Construct and use mathematical

models involving differential equations Explore learning

and population models, including exponential and logistic growth

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ReCall Mathematical Modeling1. DEVELOP MATH EQUATIONS that

represent some RealWorld Process• Almost always involves some simplifying

ASSUMPTIONS

2. SOLVE the Math Equations for the quanty/quantities of Interest

3. INTERPRET the Solution – Does it MATCH the RealWorld Results?



Differential Equations A DIFFERENTIAL EQUATION is ANY

equation that includes at least ONE calculus-type derivative• ReCall that Derivatives are themselves the

ratio “differentials” such as dy/dx or dy/dt TWO Types of Differential Equations

• ORDINARY (ODE) → Exactly ONE-Each INdependent & Dependent Variable

• PARTIAL (PDE) → Multiple Independent Variables



Differential Equation ODE Examples

• ODEs Covered in MTH16

PDE’s

• PDEs NOT covered in MTH16

2tydtdy

tkydtdyc

dtydm cos2

2

xez

dxdzz

dxzd

22

2

14 0cossin tagdtdL

tP

DrPr

rrv

v

v

11 2

2

tzw

yv

xu



Terms of the (ODE) Trade a SOLUTION to an ODE is a

FUNCTION that makes BOTH SIDES of the Original ODE TRUE at same time

A GENERAL Solution is a Characterization of a Family of Solutions• Sometimes called the Complementary

Solution



Terms of the (ODE) Trade ODEs coupled with side conditions are

called• Initial Value Problems (IVP) for a temporal

(time-based) independent variable• Boundary Value Problems (BVP) for a

spatial (distance-based) independent variable

a Solution that the satisfies the complementary eqn and side-condition is called the Particular Solution



Example Develop Model After being implanted in a mouse, the

growth rate in volume of a human colon cell over time is proportional to the difference between a maximum size M and the cell’s current volume V

Write a differential equation in terms of V, M, t, and/or a constant of proportionality that expresses this rate of change mathematically.



Example Develop Model SOLUTION: Translate the Problem Statement

Phrase-by-Phrase“…the growth rate in volume of a human

colon cell over time is proportional to the difference between a maximum size M and the cell’s current volume V…”

Build the ODE Math Model



Separation of Variables The form of a “Variable Separable”

Ordinary Differential Equation

Find The General Solution by SEPARATING THE VARIABLES and Integrating

yvtu

dtdy

ygxh

dxdy

or

dttudyyvdxxhdyyg or



Example Solve Mouse ODE Consider the differential equation for

cell growth constructed previously. • The colon cell’s maximum volume is 14

cubic millimeters• The cell’sits current volume is 0.5 cubic

millimeters• Six days later the cell has volume

increases 4 cubic millimeters. Find the Particular Solution matching

the above criteria.



Example Solve Mouse ODE SOLUTION: ReCall the ODE

Math Model From the Problem Statement,

the Maximum Volume Using M = 14 in the ODE State the

Initial Value Problem as

VMkdtdV

3mm 14M

WithTimeBased

Values

3

3

mm 46mm 5.00

tVtV



Example Solve Mouse ODE The ODE is separable, so isolate

factors that can be integrated with respect to V and those that can be integrated with respect to t

Then the Variable-Separated Equation

VkdtdV

14 VdtVk

dtdV

141

14



Example Solve Mouse ODE Integrate

Both Sidesans Solve

Ckt eAAetV where,14



Example Solve Mouse ODE At this Point have 2 Unknowns: Use the Given Time-Points (initial

values) to Generate Two Equations in Two Unknowns

Using V(0) = 0.5 mm3

kA &



Example Solve Mouse ODE Now use the other Time Point:

Thus the particular solution for the volume of the cell after t days is

3mm 46 V

05.0)27/20ln(61 k

tetV day

05.033 mm 5.13 mm 14



Example Verify ODE Solution Verify ODE↔Solution Pair

• ODE

• Solution Take Derivative of Proposed Solution

BtdtdB 21

2

2 ttetB



Example Verify ODE Solution Sub into ODE the dB/dt relation

Which by Transitive Property Suggests

Thus by Calculus and Algebra on the ODE

Which IS the ProPosed Solution for B

BtdtdB 21

dtdBte tt 212

2

teBt tt 212212

2

2 tteB



WhiteBoard Work Problems From §9.1

• P52 → Work Efficiency

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All Done for Today

GolfBallFLOW

Separation



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22















Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Documents

pe chabot college mathematics9

ode tradea solution

complementary solution

independent variablea

ode tradeodes

n s j o h n s o n c

original ode true

timea general solution