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change of the independent variable, or by some equivalent process. The general form of a linear equation of the first order is

+ Py = X,

P and X being functions of x. Assume

dt = Pdæ, so that t = [Pdx ; then the equation becomes

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where A1, A2,...A, are constants, can always be integrated by a change of the independent variable.

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where Z is what X becomes, when we substitute in it % for x. As 6", 61-, &c. are constants, this equation may, by dividing by b", be put under the form

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· In this equation make 43 = dt, or x = e. Then by Ex. 6. of Chap. 111. of the Diff. Calc. we have

so that the substitution of t for % will give rise to an equation of the form

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where T is what Z' becomes when we substitute in it er for %. The coefficients Bi, B2, &c. are constant, so that this equation is integrable by the method given in the last section. This transformation was first given by Legendre, Mémoires de l'Académie, 1787, p. 336.

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Making x = c', the transformed equation is

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Changing the independent variable from « to t, and making x = 6', this becomes

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When the independent variable is changed, the operating function is found to contain three equal factors, hence the integral is

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In other cases the reduction may be made by artifices suggested by the form of the equation.

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d' (wy) d'y dy Now

da2

" do da

dy , 2 dt 1 do (y) Therefore

da2 x dx x do . The given equation may therefore be put under the form

do (@g)

- a* (wy) = 0;

doo which is a linear equation with constant coefficients. The integral of this is evidently

xy = Ce** + Ce-ar; and therefore y=(Ce“+ C6-47) is the integral of the given equation.

(10) Let (n- ) y = 0.
This may be put under the form

1 + 3) + y = 0. Integrating with respect to x, that is, operating on both sides of the equation with c) ", we have

dr din

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