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Now the simplest process by which these equations are formed from integral equations, is the elimination of one or more constants, or of a determinate function from the equation. If the function is indeterminate, and of variables combined in a given relation as in Art. 52, the resulting differential equation will be partial and so also may it be when the function is of three or more variables. Of all these several cases instances will be given in the following Articles; and we shall first take the most simple; that viz. in which the equation arises immediately by differentiation, and the consequent elimination of one or more constants.

Since a constant quantity connected with a variable by the symbols of addition and subtraction disappears in differentiation, it follows that if an equation can be put under the form,

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from which the constant has disappeared. And therefore, whatever may have been its value in the first or primitive equation, the derived equation contains no trace of it. Similarly, if an equation can be put under the form,

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and thus in two differentiations two arbitrary constants will have disappeared; and the resulting differential equation is of the second order.

And similarly, if

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y = f(x)+cn x2 +Сn-1 x11 + ... + C2 x2 + C1 x + Co,

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and thus all the n+1 arbitrary constants have disappeared in the process of n+1 differentiations; whenever therefore the functions admit of being put under the above forms, at each

differentiation one arbitrary constant will disappear; and the resulting differential equation is of the (n+1)th order.

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whence it appears, that by one differentiation either a or b may be eliminated, and the result is a differential equation of the first order; and by two differentiations both may be made to disappear, and the differential equation is of the second order.

99.] Suppose however that the equation of relation between x and y is implicit, involves m arbitrary constants, and is of the form, u = F(x, y) = 0;

(168) for the sake of simplicity consider a to be equicrescent, and differentiate (168) n times in succession; thereby n different equations will be formed, which, added to (168), give us n +1 different equations involving m constants. Let us suppose m to be greater than n; then by means of these, n of the constants may be eliminated, and, theoretically at least, any n of them; whereby the final equation will contain m-n of the constants; and of course there may be as many different final equations as there are combinations of the m constants taken n and n together; that is, there may be

m (m −1) (m −2) (m − n + 1)

1.2.3

...

... n

different equations.

Thus suppose (168) to be differentiated twice; three equations will then be given containing m constants. By means of which two may be eliminated, and the resulting equation will contain m-2 constants; and as any two may be eliminated, we may have as many different final equations as there are combinations of the m constants taken 2 and 2 together. Hence it follows, that if we differentiate m times, there will be altogether m+1 equations, from which the m arbitrary constants may be

entirely eliminated; and that, generally, m constants involved in the primitive cannot be eliminated, unless there are formed m derived-functions of the primitive expression.

Ex. 1. y2= m (a2x2); it is required to eliminate m and a.

y2

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also eliminating m between this and the primitive, we have

(169)

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That is, we have two differential equations of the first order,

which contain

dy dx

only, and which respectively do not involve

a and m; and one differential equation of the second order, viz. (171), from which both the constants have disappeared.

Ex. 2. Eliminate the constant a from

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whence, by the common process of elimination, and dividing out the common factors,

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which is a differential equation of the first order.

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which is a differential equation of the first order.

Ex. 4. Given (x− a)2+(y—ß)2 = p2; it is required to eliminate a and ẞ by differentiation.

Let neither a nor y be equicrescent; then

(x-a) dx + (y-ẞ) dy = 0,

(x − a) d3x + (y —ẞ) d2y + dx2 + dy2 = 0 ;

.. x-a =

(dx2 + dy2) dy dx d2y - dy d2x'

therefore squaring and adding,

(x − a)2+(y—ẞ)2 = p2 =

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which is a differential expression of the second order.

100.] Also since the differentials of logarithmic and inversecircular functions are algebraical quantities; the differentials of exponential functions reproduce themselves, and of circular functions are other circular quantities related to them by trigonometrical formulæ; such transcendental functions may be eliminated from given primitive equations by means of differentiation, and differential equations will thereby be formed. The following examples will explain the process, and enable the student to apply it to other similar problems.

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Ex. 1. If y = (a2+x2)ñ; it is required to eliminate the irrational function.

m

m

y = (a2 + x2)ñ;

... log y = = log (a2+x2),

n

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dy

У

2m xy
n (a2+x2);

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a differential equation which is free from radical quantities.

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which is a differential equation of the second order.

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which is a differential equation of the first order, and contains an irrational quantity which we may remove by a subsequent differentiation. Thus

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a differential equation of the second order which is free from radicals and transcendental functions.

Ex. 4.

y = bear cos (nx+c);

dy

= abeax cos (nx+c)—nbeax sin (nx+c),

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=ay-nbeax sin (nx+c),

dy

dx

-nbaeax sin (nx + c) — n2 beax cos (nx+c),

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a differential equation of the second order, and free from exponential and circular functions.

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