View source for Differentiability of solutions (of differential equations)

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A property of solutions of differential equations, viz. that the solutions posses a specific number of continuous derivatives with respect to the independent variable  $  t $
and the parameter  $  \mu $
appearing in the equation. In the theory of differential equations the problem is posed as follows: What are the properties which the right-hand side of the equation must have for the solution to have a given number of continuous derivatives with respect to  $  t $
and  $  \mu $? 
This problem has been most thoroughly investigated for ordinary differential equations [[#References|[1]]], [[#References|[4]]].

Consider an equation of the type ( $  x $
may also be a vector):

$$ \tag{1 }

\frac{dx}{dt}
  =  f ( t , x , \mu ) ,
$$

where  $  \mu $
is a parameter (usually also a vector), and let  $  x ( t , \mu ) $
be a solution of (1) defined by the initial condition

$$ \tag{2 }
\left . x \right | _ {t = t _ {0}  }  =  x _ {0} .
$$

First differentiability of the solution with respect to  $  t $
is considered. If  $  f $
is continuous with respect to  $  t $
and  $  x $, 
the theorem on the existence of a continuous solution of the problem (1)–(2) is applicable in some domain, and then it follows from the identity which is obtained after substitution of  $  x ( t , \mu ) $
in (1) that the continuous derivative  $  x _ {t} $
also exists. The presence of  $  n $
continuous derivatives of  $  f $
with respect to  $  t $
and  $  x $
means that there exist  $  n+ 1 $
continuous derivatives of the solution with respect to  $  t $; 
$  x _ {t}  ^ {(} n) $
may be found (expressed in terms of  $  x ( t , \mu ) $) 
by successive differentiation of the identity obtained by substituting  $  x ( t , \mu ) $
in (1).

In several problems, e.g. in constructing the asymptotics of the solution in the parameter  $  \mu $, 
it is necessary to study derivatives of  $  x ( t , \mu ) $
with respect to  $  \mu $. 
In order to be specific, the existence of derivatives with respect to  $  \mu $
for  $  \mu = 0 $
will be considered. If  $  f ( t , x , \mu ) $
is continuous and has continuous partial derivatives with respect to  $  x $
and  $  \mu $
in some domain,  $  \eta _ {1} = x _  \mu  $
exists and is defined from the so-called variational equation (equation in variations, which is linear in  $  \eta _ {1} $), 
obtained from (1) by differentiating both parts with respect to  $  \mu $
and putting  $  \mu = 0 $:

$$ \tag{3 }

\frac{d \eta _ {1} }{dt}
  =  f _ {x} ( t , x ( t , 0 ) , 0 )
\eta _ {1} + f _  \mu  ( t , x ( t , 0 ) , 0 ) ,
$$

and with the aid of the initial condition

$$ \tag{4 }
\left . \eta _ {1} \right | _ {t = t _ {0}  }  =  0
$$

if  $  x _ {0} $
is independent of  $  \mu $; 
if, however,  $  x _ {0} = x _ {0} ( \mu ) $, 
then  $  \eta _ {1} \mid  _ {t = t _ {0}  } = x _ {0} ( 0) $.

The derivative  $  \eta _ {k} $
of  $  x ( t , \mu ) $
with respect to  $  \mu $
of order  $  k $(
under the condition that  $  f $
has continuous partial derivatives up to order  $  k $) 
is defined by the variational equation of the  $  k $-
th order, which differs from (3) only in its inhomogeneity, and it depends on  $  t , x ( t , 0 ) , \eta _ {1} \dots \eta _ {k-} 1 $. 
In the presence of  $  k+ 1 $
continuous derivatives of  $  x ( t , \mu ) $
with respect to  $  \mu $, 
Taylor's formula may be used as the asymptotic formula for  $  x ( t , \mu ) $
with respect to  $  \mu $:

$$ \tag{5 }
x ( t , \mu )  =  x ( t , 0 ) + \mu \eta _ {1} ( t) + \dots + \mu  ^ {k}
\eta _ {k} ( t) + O ( \mu  ^ {k+} 1 ) .
$$

This is very important, since  $  x ( t , 0 ) $
and  $  \eta _ {i} $
can then be found from equations simpler than (1).

If the right-hand side depends analytically on its arguments, the solution is an analytic function of the parameter  $  \mu $(
see, for example, [[#References|[2]]]).

The problem of differentiability of solutions with respect to  $  \mu $
is still meaningful in several cases when the right-hand side does not depend regularly on  $  \mu $. 
In one such case  $  \mu $
appears as the coefficient in front of the derivative:

$$ \tag{6 }
\mu 
\frac{dy}{dt}
  =  F ( y , x , t ) ,\  \left . y \right | _ {t = t _ {0}  }
 =  y _ {0} ,
$$

$$ 

\frac{dx}{dt}
  =  f ( y , x , t ) ,\  \left . x \right | _ {t = t _ {0}  }  =  x _ {0.}  $$

If (6) is rewritten in the form (1), i.e. is solved with respect to the derivatives, a pole-type singularity appears on the right-hand side as  $  \mu \rightarrow 0 $. 
It is found that, in the presence of  $  k+ 1 $
continuous derivatives of the right-hand sides and under certain special conditions (the so-called stability conditions), expansion (5) is valid, where  $  \eta _ {i} $
are the limit values of the derivatives with respect to  $  \mu $
of the solution of (6) as  $  \mu \rightarrow 0 $, 
which are defined by the variational equation constructed according to the same rule: (6) is differentiated with respect to  $  \mu $
and  $  \mu $
is set equal to zero. However, as distinct from the regular case, the system of variational equations will be of a lower order than (6), and the initial values for  $  \eta _ {i} $
will no longer be zero — but will be equal to (usually non-zero) constants, obtained by a definite rule [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.G. Petrovskii,   "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Tikhonov,   "On the dependence of solutions of differential equations on a small parameter"  ''Mat. Sb.'' , '''22''' :  2  (1948)  pp. 193–204  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.B. Vasil'eva,   V.F. Butuzov,   "Asymptotic expansions of solutions of singularly perturbed equations" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.N. Tikhonov,   A.B. Vasil'eva,   A.G. Sveshnikov,   "Differential equations" , Springer  (1985)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.R. Smith,   "Singular perturbation theory" , Cambridge Univ. Press  (1985)</TD></TR></table>