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Lecture 2.2: Initial value problems Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations M. Macauley (Clemson) Lecture 2.2: Initial value problems

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Lecture 2.2: Initial value problems

Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 1 / 5

SLIDE 2

Introduction

Solving an ODE generally yields an infinite family of solutions, called the general solution.

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 2 / 5

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Introduction

Solving an ODE generally yields an infinite family of solutions, called the general solution. Once we specify a point (t0, y(t0)), we determine a particular solution.

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 2 / 5

SLIDE 4

Introduction

Solving an ODE generally yields an infinite family of solutions, called the general solution. Once we specify a point (t0, y(t0)), we determine a particular solution.

Definition

An ODE with a specified point y(t0) = y0 is called an initial value problem (IVP).

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 2 / 5

SLIDE 5

Introduction

Solving an ODE generally yields an infinite family of solutions, called the general solution. Once we specify a point (t0, y(t0)), we determine a particular solution.

Definition

An ODE with a specified point y(t0) = y0 is called an initial value problem (IVP).

Example 1

Solve the IVP y ′ = k(72 − y); y(0) = 90.

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 2 / 5

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Word problems

Example 2 (exponential growth)

A house sells in 2003 for $179,500 and was on sale in 2008 for $319,500. (a) What was the average rate of appreciation of the value? (b) Suppose the market has been increasing at a 9% rate. How much is the house worth?

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 3 / 5

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Word problems

Example 3 (exponential decay)

You have 10 grams of a radioactive substance. Three years later, you have 4 grams. (a) What is the half-life? (b) How long until only 1 gram remains?

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 4 / 5

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Word problems

Example 3 (exponential decay)

You have 10 grams of a radioactive substance. Three years later, you have 4 grams. (a) What is the half-life? (b) How long until only 1 gram remains?

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 4 / 5

SLIDE 9

Word problems

Example 4 (exponential decay to a value)

My coffee is 120◦ when class starts, and the classroom is 75◦. After 30 minutes, the coffee is 100◦. (a) What will the temperature be at the end of class (t = 50)? (b) Suppose it was brewed at 160◦. When did I brew it?

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 5 / 5

SLIDE 10

Word problems

Example 4 (exponential decay to a value)

My coffee is 120◦ when class starts, and the classroom is 75◦. After 30 minutes, the coffee is 100◦. (a) What will the temperature be at the end of class (t = 50)? (b) Suppose it was brewed at 160◦. When did I brew it?

M. Macauley (Clemson)

Lecture 2.2: Initial value problems Math 2080, ODEs 5 / 5