In other words, how does the derivative function relate to the original function? This isn't so much a mathematical question as a conceptual question. It may not help me pass exams, but I want to understand.
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We represent a single curve by a Cartesian equation which is algebraic.
For example, the equation of a circle having O(0, 0) as the center and radius = 5 is given by
x^2 + y^2 = 25.
But, suppose we want to represent all the equations of circles having center O(0, 0) having any radius, r, it will be
x^2 + y^2 = r^2.
We call this equation as the equation of a family of cirlces having center (0, 0) and radius r, where r has any positive real value.
If we differentiate the above equation, we get
2x + 2y dy/dx = 0
=> dy/dx = - (x/y)
This is a differential equation of first order and it also represents the family of circles having center (0, 0) and any radius r. Note that the equation does not contain r.
Taking another example,
y = mx + b represents a family of straight lines having slope m and y-intercept = b where m and b can be any real numbers. Assigning different sets of values to m and b, we get all the lines of the family. Note that as m has to be defined, this family of lines include all possible lines in the coordinate plane except the vertical lines whose slopes are not defined.
Differentiating, y = mx + b
=> dy/dx = m
and d^2y/dx^2 = 0
This is a differential equation of the second order representing all possible lines except the vertical lines. Note again that m and b are not there in the equation.
Summarising, a Cartesian equation which contains one or more constants whose values can be varied represents a family of curves The constant is called an arbitrary constant. By differentiation, we can get a differential equation representing the same family of curves and the differential equation is free from the constant. If there is one arbitrary constant, differential equation will be of the first order and if there are 2 or 3 or n arbitrary constants, the differential equation will be of second, third or nth order. The order of the differential equation equivalent to the Cartesian equation has to be exactly the same as the number of arbitrary constants, no more and no less. The differential equation should not contain the arbitrary constant. Often, it is a challenging exercise to convert the Cartesian equation with 2 or more arbitrary constants into its equaivalent differential equation.
Edit:
To understand the challenge involved in forming the differential equation which represents a given family of curves, refer to my following answer posted in my blog.
http://schoolnotes4u.blogspot.in/2012/07…
For example, the equation of a circle having O(0, 0) as the center and radius = 5 is given by
x^2 + y^2 = 25.
But, suppose we want to represent all the equations of circles having center O(0, 0) having any radius, r, it will be
x^2 + y^2 = r^2.
We call this equation as the equation of a family of cirlces having center (0, 0) and radius r, where r has any positive real value.
If we differentiate the above equation, we get
2x + 2y dy/dx = 0
=> dy/dx = - (x/y)
This is a differential equation of first order and it also represents the family of circles having center (0, 0) and any radius r. Note that the equation does not contain r.
Taking another example,
y = mx + b represents a family of straight lines having slope m and y-intercept = b where m and b can be any real numbers. Assigning different sets of values to m and b, we get all the lines of the family. Note that as m has to be defined, this family of lines include all possible lines in the coordinate plane except the vertical lines whose slopes are not defined.
Differentiating, y = mx + b
=> dy/dx = m
and d^2y/dx^2 = 0
This is a differential equation of the second order representing all possible lines except the vertical lines. Note again that m and b are not there in the equation.
Summarising, a Cartesian equation which contains one or more constants whose values can be varied represents a family of curves The constant is called an arbitrary constant. By differentiation, we can get a differential equation representing the same family of curves and the differential equation is free from the constant. If there is one arbitrary constant, differential equation will be of the first order and if there are 2 or 3 or n arbitrary constants, the differential equation will be of second, third or nth order. The order of the differential equation equivalent to the Cartesian equation has to be exactly the same as the number of arbitrary constants, no more and no less. The differential equation should not contain the arbitrary constant. Often, it is a challenging exercise to convert the Cartesian equation with 2 or more arbitrary constants into its equaivalent differential equation.
Edit:
To understand the challenge involved in forming the differential equation which represents a given family of curves, refer to my following answer posted in my blog.
http://schoolnotes4u.blogspot.in/2012/07…
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The differential part means the slope or gradient of a curve. A differential equation is one that links the gradient of a curve with the value of a curve. EG for a bucket with a hole in it, the rate of change of the water level (the differential bit) is dependant on the actual water level.
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Very good question.
I found this good website
http://mathforum.org/library/drmath/view/64456.html
I found this good website
http://mathforum.org/library/drmath/view/64456.html