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An ordinary differential equation which has at least one [[Oscillating solution|oscillating solution]]. There are different concepts of the oscillation of a solution. The most widespread are oscillation at a point (usually taken to be  $  + \infty $) 
and oscillation on an interval. A non-zero solution of the equation

$$ \tag{1 }
u  ^ {(} n)  =  f( t, u , u  ^  \prime  \dots u  ^ {(} n- 1) ),\  n \geq  2 ,
$$

where  $  f( t, 0 \dots 0) = 0 $, 
is called oscillating at the point  $  + \infty $(
or on an interval  $  I  $) 
if it has a sequence of zeros which converges to  $  + \infty $(
respectively, there are at least  $  n $
zeros in  $  I $
counted according to their multiplicity). Equation (1) is oscillating at  $  + \infty $
or on an interval  $  I $
if its solutions are oscillating (at  $  + \infty $, 
respectively, on  $  I  $).

Among equations which are oscillatory at  $  + \infty $
the equations which possess the properties  $  A $
or  $  B $, 
i.e. which are compatible in a specific sense with one of the equations

$$ 
u  ^ {(} n)  =  - u \  \textrm{ or } \  u  ^ {(} n)  =  u ,
$$

are distinguished. Equation (1) is said to possess property  $  A $
if all its solutions defined in a neighbourhood of  $  + \infty $
are oscillating when  $  n $
is even; when  $  n $
is odd, they should either be oscillating or satisfy the condition

$$ \tag{2 }
\lim\limits _ {t \rightarrow + \infty }  u  ^ {(} i- 1) ( t)  =  0,\  i = 1 \dots n.
$$

If every solution of equation (1) defined in a neighbourhood of  $  + \infty $, 
when  $  n $
is even, is either oscillating, or satisfies condition (2) or

$$ \tag{3 }
\lim\limits _ {t \rightarrow + \infty }  | u  ^ {(} i- 1) ( t) |  =  + \infty ,\ \ 
i = 1 \dots n,
$$

while when  $  n $
is odd, it is either oscillating or satisfies condition (3), then the equation possesses property  $  B $.

The linear equation

$$ \tag{4 }
u  ^ {(} n)  =  a( t) u
$$

with a locally summable coefficient  $  a:  [ t _ {0} , + \infty ) \rightarrow \mathbf R $
possesses property  $  A $(
property  $  B $) 
if

$$ 
a( t)  \leq   0 \  ( a( t)  \geq   0) \  \textrm{ when }  t \geq  t _ {0}  $$

and either

$$ 
\int\limits _ {t _ {0} } ^ { {+ }  \infty } t ^ {n- 1- \epsilon } | a( t) |  dt  =  + \infty
$$

or

$$ 
a( t)  \leq   
\frac{\mu _ {n} - \epsilon }{t  ^ {n} }
 \  \left ( a( t)  \geq   
\frac{\nu _ {n} + \epsilon }{t  ^ {n} }
 \right )
$$

when  $  t \geq  t _ {0} $, 
where  $  \epsilon > 0 $
and  $  \mu _ {n} $
is the smallest ( $  \nu _ {n} $
is the largest) of the local minima (maxima) of the polynomial  $  x( x- 1) \dots ( x- n+ 1) $(
see [[#References|[1]]]–[[#References|[5]]]).

An equation of Emden–Fowler type

$$ \tag{5 }
u  ^ {(} n)  =  a( t) | u |  ^  \lambda    \mathop{\rm sign}  u ,\ \ 
\lambda > 0,\  \lambda \neq 1,
$$

with a locally summable non-positive (non-negative) coefficient  $  a:  [ t _ {0} , + \infty ) \rightarrow \mathbf R $
possesses property  $  A $(
property  $  B $) 
if and only if

$$ 
\int\limits _ {t _ {0} } ^ { {+ }  \infty } t  ^  \mu  | a( t) |  dt  =  + \infty ,
$$

where  $  \mu = \min \{ n- 1, ( n- 1) \lambda \} $(
see [[#References|[4]]], [[#References|[6]]], [[#References|[7]]]).

In a number of cases the question of the oscillation of equation (1) can be reduced to the same question for the standard equations of the form (4) and (5) using a [[Comparison theorem|comparison theorem]] (see [[#References|[11]]]).

In studying the oscillatory properties of equations with deviating argument, certain specific features arise. For example, if  $  n $
is odd,  $  \Delta > 0 $, 
and if for large  $  t $
the inequality

$$ 
a( t)  \leq   a _ {0}  <  - n! \Delta  ^ {-} n
$$

is fulfilled, then all non-zero solutions of the equation

$$ 
u  ^ {(} n) ( t)  =  a( t) u( t - \Delta )
$$

are oscillatory at  $  + \infty $(
see [[#References|[10]]], [[#References|[11]]]). At the same time, if  $  a $
is non-positive and  $  n $
is odd, the non-retarded equation (4) always has a non-oscillating solution.

The concepts of oscillation and non-oscillation on an interval are generally studied for linear homogeneous equations. They are of fundamental value in the theory of boundary value problems (see [[#References|[12]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Kneser,   "Untersuchungen über die reellen Nullstellen der Integrale linearer Integralgleichungen"  ''Math. Ann.'' , '''42'''  (1893)  pp. 409–435</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.G. Mikusinksi,   "On Fite's oscillation theorems"  ''Colloq. Math.'' , '''2'''  (1951)  pp. 34–39</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Kondrat'ev,   "The oscillatory character of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047056.png" />"  ''Trudy Moskov. Mat. Obshch.'' , '''10'''  (1961)  pp. 419–436  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.T. Kiguradze,   "On the oscillatory character of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047057.png" />"  ''Mat. Sb.'' , '''65''' :  2  (1964)  pp. 172–187  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T.A. Chanturiya,   "On a comparison theorem for linear differential equations"  ''Math. USSR Izv.'' , '''10''' :  5  (1976)  pp. 1075–1088  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''40''' :  5  (1976)  pp. 1128–1142</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I. Ličko,   M. Švec,   "La charactère oscillatoire des solutions de l'équation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047059.png" />"  ''Chekhosl. Mat. Zh.'' , '''13'''  (1963)  pp. 481–491</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I.T. Kiguradze,   "On the oscillatory and monotone solutions of ordinary differential equations"  ''Arch. Math.'' , '''14''' :  1  (1978)  pp. 21–44</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  C.A. Swanson,   "Comparison and oscillation theory of linear differential equations" , Acad. Press  (1968)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.D. Myshkis,   "Linear differential equations with retarded argument" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R.G. Koplatadze,   T.A. Chanturiya,   "On the oscillatory properties of differential equations with deviating argument" , Tbilisi  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  A.Yu. Levin,   "Non-oscillation of the solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070470/o07047060.png" />"  ''Russian Math. Surveys'' , '''24''' :  2  (1969)  pp. 43–99  ''Uspekhi Mat. Nauk'' , '''24''' :  2  (1969)  pp. 43–96</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.T. Reid,   "Sturmian theory for ordinary differential equations" , Springer  (1980)</TD></TR></table>