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Growth of solutions of second order linear differential equations. (English) Zbl 1151.34069

The authors study the growth of solutions of the linear differential equation

f '' +A 1 (z)e az f ' +A 0 (z)e bz f=H(z),(2)

where A 1 (z), A 0 (z) and H(z) are entire functions of order less than one, and a,b.

The authors prove the following theorems

Theorem 1.1. Suppose that A 0 ¬0, A 1 ¬0, H are entire functions of order less than one, and the complex constants a, b satisfy ab0 and ab. Then every nontrivial solution f of (2) is of infinite order.

Theorem 1.3. Suppose that A 0 ¬0, A 1 ¬0, D 0 , D 1 , H are entire functions of order less than one, and the complex constants a, b satisfy ab0 and b/a<1. Then every nontrivial solution f of equation

f '' +(A 1 (z)e a·z +D 1 (z))f ' +(A 0 (z)e b·z +D 0 (z))f=H(z)

is of infinite order.


MSC:
34M10Oscillation, growth of solutions (ODE in the complex domain)
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