  We introduce the notion of conditional Lipschitz shadowing, which does not
aim to shadow every pseudo-orbit, but only those which belong to a certain
prescribed set. We establish two types of sufficient conditions under which
certain non\-auto\-nomous ordinary differential equations have such a property.
The first criterion applies to a semilinear differential equation provided that
its linear part is hyperbolic and the nonlinearity is small in a neighborhood
of the prescribed set. The second criterion requires that the logarithmic norm
of the derivative of the right-hand side with respect to the state variable is
uniformly negative in a neighborhood of the prescribed set. The results are
applicable to important classes of model equations including the logistic
equation, whose conditional shadowing has recently been studied. Several
examples are constructed showing that the obtained conditions are optimal.
