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Two theorems are presented which guarantee the boundedness and oscillation of solutions of certain classes of second order nonlinear differential equations. In this paper we shall show boundedness of solutions of the differential equation y"+a(t)b(y)=0 with appropriate conditions on a(t) and b(y). Furthermore, all solutions will be oscillatory subject to a further condition. By oscillatory we shall mean a function having arbitrarily large zeros. The first theorem is an extension of a result of Utz [1] in which he considers b(y)=y2n~1. We now prove a result for boundedness of solutions. Theorem I. If a(t)> a>0 and da/dt^O on [T, oo), b(y) continuous, and limj,^±00 B(y)=$yb(u) du=co then all solutions of (1) / + a(t)b(y)=0 are bounded as r—>-oo. Proof. Multiply (1) by 2/ to get (2) 2y'y" + 2a(t)b(y)y' = 0. Upon integration by parts we have (3) (y'(uf)\l + 2a(u)B(y(u))\\o 2 P df B(y) du = 0. J ir, Uli Thus (4) y'(tf + 2a(t)B(y(t)) 2 f ^ B(y) du = K Jto du where K=y'(t0)2-r-2a(to)B(y(to)). Now the above implies \y'\ and \y\ remain bounded. If not, the left side of (4) would become infinite which is impossible. All solutions of (1) will be oscillatory if sgn j=sgn b(y). To prove this we need the following lemma used by Utz [2]. Received by the editors September 21,1971 and, in revised form, December 17, 1971. AMS 1970 subject classifications. Primary 34C10, 34C15.

Properties of solutions of differential equations of the form $y\sp”+a(t)b(y)=0$