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By Goursat E.

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In fact, we have Wl(XO) = w~(xo) = 0, w~(xo) = - 1, W2(XO) = w~(xo) = 0, wHxo) = -1, W3(XO) = -1, w;(xo) = 0, w~(xo) = 2A(xo). 10, Wl(X)xo. Hence Third Order, Equations in Normal Form 43 Y3(X) is also oscillatory in (xo, 00), because it belongs to a regular band with the solution Yz, which is oscillatory. The general solution of the differential equation (a) is Y = ClYl+ CzYz + C3Y3. We have y(xo) = C3, y'(xo) = C2, y"(xo) = Cl. If at least one of the constants ChCZ, C3 wanishes, then the solution y is oscillatory.

2 lim+ [z' - z x_a y,] = 0. y The situation at {3 - is similar. 9). 3) are identical for a ~ x ~ {3 and y and z are dependent, then A == At. K = 0, b == 0 for a ~ x ~ {3 and Z' Y - ZY' = 0, therefore Y = cZ, where c is a suitable constant. Thus the theorem is proved. 2. e. at least one solution has infinitely many zeros in (a, b). 4), a < xo< b, oscillates in (a, b). 3. 4) have oscillatory solutions in (a, b), a < a < b. e. 4)) is oscillatory in (a, b). Besides the differential equation (a), let us consider the following differential equation z"'+2A 1z'+(A;+bl )z=0, (al) where Al = AI(x), A; = A;(x), b l xe(a, b).

PROOF. 2. Then evidently (~)' = vu' ~2 uv' Integration of the last equality from al to a2 gives _l 0- a2 a, vu' --2 uv' dt. V This can only hold if the function vu' - flv' has at least one zero between al and a2 at which it changes its sign. Denote such a zero by T. Then, at T, the equations 19 Third Order Equations in Normal Form Clu(r)-C2 v (r)=0, CIU'(r) + C2V'(r) = ° have a non-trivial solution for Ct, C2, whence the assertion follows by putting A. = u(r)/v(r). 0 Suppose, further, that the Laguerre invariant has the property b(x)~O for xE(a, b).