# [MATH 4B] - Midterm Exam Guide - Comprehensive Notes for the exam (16 pages long!)

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UCSB

MATH 4B

MIDTERM EXAM

STUDY GUIDE

Order of

is 1

Order of ODE = the order of the highest derivative that appears

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System of ODEs = collection of several ODEs involving a number of unknown functions

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ODE = equation involving an unknown function of a single variable and its derivatives up to some

fixed order

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Partial Differential Equation = equation that involves an unknown function of more than one

variable and its (partial) derivatives

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What is an Ordinary Differential Equation?

Goal: to understand physical systems that evolve over time, both qualitatively + quantitatively

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Physical laws are often stated in terms of rates of change --> derivatives

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Why Study ODEs?

"y is a function whose derivative is equal to x"

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Trying to find y

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Solution:

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with

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Sometimes we have initial conditions that allow us to determine the arbitrary constant in our

solution:

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Suppose is a continuous function

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Ie.

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Examples of ODEs

-->

-->

-->

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The most general ODE of first order has the form:

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General First Order ODEs

for all

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A function defined on some interval is a solution of the ODE if:

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What is a Solution of an ODE?

Linear in terms of y

First order ODE is linear if there are functions and such that

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First Order Linear ODEs

Lecture 1

Tuesday, January 10, 2017

2:00 PM

MATH 4B Page 1

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Linear in terms of y

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The function defines a slope/direction field

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Geometric Interpretation of ODEs -- Direction Fields

Lecture 1

MATH 4B Page 2

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