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and each of these = ∞, if y = x; in which case c = x; that is, the arbitrary constant receives a variable value, and therefore we have a singular solution.

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The same singular solution may also be found by the methods. previously investigated. The differential equation of which the given equation is the integral is

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and as this relation satisfies the differential equation given above, it is a singular solution.

Also if we find the c-differential of the general integral, and then eliminate c according to the method of Art. 409, we have

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and thus all the methods for finding singular solutions lead to the same result.

SECTION 8.-Differential equations of the first order and of any degree.

411.] Order of differential equation depends on the index of the symbol of differentiation with which the highest differential

or differential coefficient is affected, and degree on the power to which such highest differential or differential coefficient is raised. Thus a differential expression of the first order and nth degree is that which involves (d), but no higher derived-function, and

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dy dx

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+F1 (dy)" +...+Fn-2 dx

dx

+F, 0, (214)

where F1, F2, ... F, are symbols for functions of x and y.

Let us suppose the equation (214) to be resolved into n factors

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where f1, f2,...ƒ, are the roots of (214) and are generally functions of x and y: let each of these equations be integrated separately, and let their integrals be

1(x,y,c1) = 0, 2(x,y, c2) = 0, 0,... where C1, C2, ... c, are arbitrary constants.

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n(x,y,c1) = 0, (216) Then the equation

(217)

$1 (x, y, c1) × Þ2 (X, Y, C2) × ... ×Þ„ (x, y, c2) = 0 will contain all the integrals of (214), because it and (214) vanish simultaneously for each of the n functions. And the truth of this final equation will not be affected if the arbitrary constants are equal, that is, if c1 = C2 = = Cnc, because c is arbitrary, and therefore will pass through the values C1, C2, ... C, if it receives all the values of which it is capable.

...

The following are examples of this mode of integration.

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and these two solutions may be combined into the single equation

(y—ax2-c1) (y+ax2—c2) = 0;

either of which factors satisfies the given differential equation; and if c1 = c2 = c, we have

(y-c) 2-a2 x4 = 0.

Now this is equally true with the former equation, as the primitive from which the differential equation is derived; for it may

be derived either from this latter by the elimination of c, or from either of the former by the elimination of c1 or C2.

The singular solution of this equation is x=0; and considered geometrically the general integral represents two parabolas which have a common axis, viz. that of y, and a common vertex on the axis of y at a distance c from the origin; and the singular solution represents a point on the axis of y, through which all the parabolas pass, and which is consequently an envelope.

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and which may be simplified if c1 = c2 = C2 = c. In this case also the singular solution is a point on the axis of y.

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which are homogeneous, and may be integrated by the methods

explained above.

412.] Certain forms of differential equations of the first order and of any degree are capable of integration without resolution into factors according to the method of the preceding Article, and these we proceed to explain.

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Let this equation be divided by (y); so that making obvious substitutions it becomes

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and let us in the first place take the simple case in which

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this form is known by the name of Clairaut's equation.
Let this be differentiated; then we have, resubstituting,

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(220)

(221)

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which is the general integral, containing the arbitrary constant c. We might of course integrate (222) immediately; whereby we have

y = cx + c1,

(223)

where c1 is a new arbitrary constant: but as (223) is to satisfy C1 (220), c1 = f(c). This result is also manifest from the fact that (220) is a differential equation of the first order, and therefore its integral must contain only one arbitrary constant.

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expression results which of course satisfies the differential equation, and is independent of c the arbitrary constant, and is therefore either a particular integral or a singular solution; and it is

manifestly the latter, because c which is equal to by a function of x, viz. 4(x).

dy

dx

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is replaced

The following are differential equations solved by this process, which generally is called Integration by means of Differentiation.

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and this is the singular solution, since it involves no arbitrary

constant, and is not a particular integral, because the constant is

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