This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1900 Excerpt: ...+c + /3z, subject to the condition assigned. As the equation can be integrated by quadratures, it is possible to compare this result with the complete primitive. Let. a2 dn zw--w, = r, 2) az after substitution, the equation for Ij is so that ez cz r, =Aea' + Bea See also Konigsberger, Theorie d. Differentialgleichungen ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1900 Excerpt: ...+c + /3z, subject to the condition assigned. As the equation can be integrated by quadratures, it is possible to compare this result with the complete primitive. Let. a2 dn zw--w, = r, 2) az after substitution, the equation for Ij is so that ez cz r, =Aea' + Bea See also Konigsberger, Theorie d. Differentialgleichungen, pp. 411--416. Ae"-Bdz icz z2' Aea2+B which is the complete primitive; it contains two arbitrary constants, viz. C and the fraction B/A. Denoting this fraction by k, we have dz z2 so that I_i 4f (i w--c j-. 22 1S z + a regular function of z, this regular function vanishing with z. If initially w=c, then k = 0 and the term in zlogz disappears; if initially w=-c, then k = ao and the term in zlogz again disappears; for any other assigned initial value of w, the term in zlogz remains. Non-regular Integrals Of W" = f(w', w, z). 213. The simplest deviation from the previous conditions arises, when the initial value assigned to w' is infinite. In that case, we make z the dependent variable and w the independent variable, so that W" =-Z," w z and the equations are dw j The initial conditions now are that z = c, and a' = 0, when w--a. If z' = 0, z = c, w = a, constitute an ordinary combination for the function g, the preceding existence-theorem applies: and the solution is z-c=(w-afg(0, c, a)+..., so that, if g (0, c, o) is different from zero, we have w-a = (z-cyP(z-c)i, where P is a regular function of its argument, and does not vanish when z = c: and if g (0, c, a) is zero, we have a result of the form I I w-a = (z-c)n R (z-c)"j, where n 2, and J? is a regular function of its argument, which does not vanish when z = c. But when z' is an integral factor of g, so that-, g (z', z, w) is regular in the vicinity of ts z = 0, z = c, ...
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