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| Formation of a Differential Equation |
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| Consider the family of lines represented by y = mx ….(1) |
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| This equation represents infinite number of lines passing through the origin. |
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| Differentiating (1), we get |
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 |
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| Substituting this value of m, we get the differential equation. |
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 |
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| Consider the family of lines represented by |
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| y = mx + c ….(3) |
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| where m and c are arbitrary constants. Any line on the co-ordinate plane can be represented by (3) |
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| Let us form a differential equation for equation (3) |
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| Differentiating equation (3) , we have |
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| Differentiating again, we have |
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| This is the differential equation which represents the family of straight lines y = mx +c. |
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| Note: |
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| The equation y = mx has one arbitrary constant and its differential equation is of order 1. The equation y = mx + c has two arbitrary constants and its differential equation is of order 2. |
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| In general, if an equation contains n arbitrary constants, then we obtain its differential equation which is of order n, after eliminatory all the n constants. |
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| Note that equation(1) is called the primitive of Differentiating equation(2). |
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| Equation (3) is called the primitive of Differentiating equation (4). |
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| The formation of a differential equation may be done by differentiating and eliminating arbitrary constants from the given equation. |
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| The given equation is differentiated as many times as there are arbitrary constants. |
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| In this section we explained with examples how to form Differential Equation. |
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| Suppose a family of curves depending on one constant is given by |
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| F1 : f (x, y, a)= 0 ….(1) |
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where a  R is the parameter |
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| Differentiating (1) with respect to x, we have |
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| g (x, y, y', a) = 0 ….(2) |
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| Now eliminating 'a' from equation (1) and equation (2), we get the required differential equation. |
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| f (x, y, y') = 0 ….(3) |
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| This equation represents the family of curves F1. Equation (1) is called the primitive of the differential equation (3). |
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| In this section, we discuss in general how to form Differential Equation which represent a family of curves. |
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| Let |
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| F2: f (x, y, a, b) = 0 ….(4) |
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where a, b R |
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| represents a family of curves which depend on two constants (parameters) a, b. |
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| Differentiating (4) with respect to x, we have |
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| g (x, y, y', a, b) = 0 ….(5) |
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| We can not eliminate a and b from equation (4) and equation (5). Therefore we need another equation which can be obtained by differentiation equation (5). |
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| Differentiating equation (5) with respect to x, we have |
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| h(x, y, y ',y'', a, b) = 0 ….(6) |
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| Now the arbitrary constants can be eliminated from equation (4), (5) and (6), to obtain the differential equation of the family of curves. |
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| As discussed earlier, the family of curves containing one parameter, is represented by a differential equation of order 1 . |
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| The family of curves which depend on two parameter is represented by differential equation of order 2. |
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| In general, the family of curves which depend on n parameter is represented by differential equation of order 3. |
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| Example 1: |
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| Form a differential equations by eliminating 'a' from the family of curves y2=4ax. |
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| Suggested answer: |
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| y2 = 4ax …(1) |
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| Differentiating with respect to x |
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| Substitute for 4a in (1), we get |
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| y - 2xy' = 0 |
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| Note that the given equation is differentiated only once to obtain the differential equation since it has only one constant. |
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| Example 2: |
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| Form a differential equation by eliminatory the parameter A and B from the family of curves given by y = Ae2x+Be-2x. |
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| Suggested answer: |
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| The given equation has two arbitrary constants. |
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| \ To obtain the differential equation, we differentiate the given equation twice. |
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| Differentiate with respect to x. |
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| y2 = 4y |
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| y2 - 4y = 0 which is the required differential equation. |
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