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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
where A1, A2,...A, are constants, can always be integrated by a change of the independent variable.
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
· 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
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.
Making x = c', the transformed equation is
Changing the independent variable from « to t, and making x = 6', this becomes
When the independent variable is changed, the operating function is found to contain three equal factors, hence the integral is
In other cases the reduction may be made by artifices suggested by the form of the equation.
d' (wy) d'y dy Now
" do da
dy , 2 dt 1 do (y) Therefore
da2 x dx x do . The given equation may therefore be put under the form
- 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.
1 + 3) + y = 0. Integrating with respect to x, that is, operating on both sides of the equation with c) ", we have