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Ordinary Differential Equations

Where do ODEs arise?
Notation and Definitions
Solution methods for 1^st order ODEs

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Where do ODE’s arise

All branches of Engineering
Economics
Biology and Medicine
Chemistry, Physics etc

Anytime you wish to find out how something changes with time (and sometimes space)

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Example – Newton’s Law of Cooling

This is a model of how the temperature of an object changes as it loses heat to the surrounding atmosphere:

Temperature of the object:

Room Temperature:

Newton’s laws states: “The rate of change in the temperature of an object is proportional to the difference in temperature between the object and the room temperature”

Form ODE

Solve

ODE

Where is the initial temperature of the object.

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Notation and Definitions

Order
Linearity
Homogeneity
Initial Value/Boundary value problems

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Order

The order of a differential equation is just the order of highest derivative used.

2^nd order

3^rd order

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Linearity

The important issue is how the unknown y appears in the equation. A linear equation involves the dependent variable (y) and its derivatives by themselves. There must be no "unusual" nonlinear functions of y or its derivatives.
A linear equation must have constant coefficients, or coefficients which depend on the independent variable (t). If y or its derivatives appear in the coefficient the equation is non-linear.

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Linearity - Examples

is linear

is non-linear

is linear

is non-linear

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Linearity – Summary

Linear	Non-linear

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Linearity – Special Property

If a linear homogeneous ODE has solutions:

and

then:

where a and b are constants,

is also a solution.

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Linearity – Special Property

Example:

has solutions

and

Check

Therefore

is also a solution:

Check

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Homogeniety

Put all the terms of the equation which involve the dependent variable on the LHS.
Homogeneous: If there is nothing left on the RHS the equation is homogeneous (unforced or free)
Nonhomogeneous: If there are terms involving t (or constants) - but not y - left on the RHS the equation is nonhomogeneous (forced)

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Example

1st order
Linear
Nonhomogeneous
Initial value problem

2nd order
Linear
Nonhomogeneous
Boundary value problem

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Example

2nd order
Nonlinear
Homogeneous
Initial value problem

2nd order
Linear
Homogeneous
Initial value problem

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Solution Methods - Direct Integration

This method works for equations where the RHS does not depend on the unknown:
The general form is:

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Direct Integration

y is called the unknown or dependent variable;
t is called the independent variable;
“solving” means finding a formula for y as a function of t;
Mostly we use t for time as the independent variable but in some cases we use x for distance.

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Direct Integration – Example

Find the velocity of a car that is accelerating from rest at 3 ms^-2:

If the car was initially at rest we have the condition:

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Solution Methods - Separation

The separation method applies only to 1^st order ODEs. It can be used if the RHS can be factored into a function of t multiplied by a function of y:

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Separation – General Idea

First Separate:

Then integrate LHS with respect to y, RHS with respect to t.

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Separation - Example

Separate:

Now integrate:

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