Differential equations are an powerful mathematical tool that help us understand nature and finance, allowing us to make accurate calculations, including:
movement of an object
the collision of two cars
trajectories of planets
As an example:
Imagine a particle that is projected horizontally (Gravity is neglected):
Velocity is given by
Acceleration is
It is assumed that the drag is proportional to
Applying Newton's second Law: "Force = Mass x Acceleration"
Therefore if we are interested in the distancex
or
If we are interested in time then:
Definitions
Particular solutions
When
and
When
and
Differential Equations which involve only one independent variable are calledOrdinary. In these equationsx is the independent variable andy is the dependent variable.For example :
Equations which involve two or more independent variables and partial differential coefficients with respect to them are calledPartial. For example :
When
The Laplace equation
Order
Equations that involve a second differential coefficient but none of higher orders is said to beSecond Order.
When
and
When
and
When
For example :
First order
Second order
Third order
Degree
The degree of an equation is the power of the highest differential coefficient once the equation has been made rational and integral as far as the differential coefficients are concerned. For example :
When
and
When
First degree
Second degree
Note that this definition of degree does not require x or y to occur rationally or integrally.
The Geometrical Meaning Of A Differential Equation
This section presents geometric characteristics of the solution of a differential equation .
Linear Solution
Alinear function is a function such as where
If then
Quadric Solution
Aquadric function is a function such as where
If then
Now consider the equationTherefore
If A = 0 the graph is as above.
If A = 1
If A = -2
Exponential Solution
Anexponential function is a function such as where
Now consider the following equation:
This can be rearranged as:The variables have now been separated and :
From which the explicit form is given by:
The Formation Of Differential Equations By Elimination
If from the following equation we eliminate the arbitrary constant we get the following:
Extending this concept, if we started withn arbitrary constants, we could eliminate them byn differentiations. The result would be a differential equation of the order.Conversley if we are given a differential equation of the order we can, in general, obtain an equivalent relationship containing no derivatives butn arbitrary constants. This relationship is called "The General Solution"For Example
wherew i aconstantIntegrating with respect tox gives
And so on until
WhereA,B,C andE are all arbitrary constants
The Complete Primitive; Particular Integral; And Singular Solution
The solution of a differential equation containing the full number of arbitrary constants is called "The Complete Primative". Any solution derived from the complete Primitive by giving particular values to these constants is called a "A Particular Integral"For example
A Particular solution of
is given by
(Obtained by putting A,B,C,E = 0)or
Example:
Example - The use of Differential Equations to Solve Problems in Dynamics
Problem
A cricket ball is thrown vertically upwards with a velocity of v ft/sec. The retardation is or . Find the maximum height reached (Y) and the time of flight to the vertex (T).Prove that the Initial velocity u is given by:
Workings
The acceleration = -kv - gTo find the time of Flight T
thus
When t = 0 v = u
Thus
At the vertex t = T and v = 0 so
i.e.
For Height Y
But when y = 0, v = u so
At the Vertex v = 0
Solution
The flight time is:
Max Height is:
Differential Equations Which Include Trigonometrical Functions The Right Hand Side
In the following worked examples is usually re-written as. For those unused to this type of trigonometricalmanipulation, the following notes should help.
Example:
Example - Basic examples .
Problem
Basic trigonometrical examples
Workings
The reference page on Trigonometrical Formulae includes:-
we can compare the right hand side with the right hand side of (5) and we can see that they are ofthe same form but has been replaced by "3" and by "4". Clearly thiscan not be correct as the Sine and Cosine can not have a value above unity but if we draw thefollowing right angled triangle.
Values of Sine and Cosine can be obtained which can be put into equation (5)
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Therefore if we are interested in the distancex
or
If we are interested in time then:
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and
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Equations which involve two or more independent variables and partial differential coefficients with respect to them are calledPartial. For example :
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Third order
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where
then
Therefore
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This can be rearranged as:
The variables have now been separated and :
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From which the explicit form is given by:
is given by
(Obtained by putting A,B,C,E = 0)or
or
. Find the maximum height reached (Y) and the time of flight to the vertex (T).Prove that the Initial velocity u is given by:
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has been replaced by "3" and
by "4". Clearly thiscan not be correct as the Sine and Cosine can not have a value above unity but if we draw thefollowing right angled triangle.
can be obtained which can be put into equation (5)This can be re-arranged to satisfy the requirements of equation (6)
