Impossible Kinds Of Differential Equations Today my professor told me that there are some differential equations that cannot be solved. Is this true? If it is true, why can they not be solved? How complex would that kind of differential equation have to be?
 A: Ordinary Differential Equations generally admit solutions in the event that the defining functions are "reasonable", i.e. possessed of Lipschitz continuity.  Differentiable functions are generally locally Lipschitz, so we know that equations of the form
$\dot {\vec x} = \vec f(\vec x, t) \tag{1}$
where $\vec x \in \Bbb R^n$, with differentiable $\vec f$, have unique solutions when the initial data
$\vec x(t_0) = \vec x_0 \tag{2}$
is specified.  Most equations of practical interest, say in the sciences or engineering, fall into this category so there's really no problem in the applications.
Partial Differential Equations, on the other hand, yield us no such good fortune.  There are even very simple, first order PDEs which admit no solutions whatsoever.  See, for instance, Lewy's example.  Lewy showed that even such a simple PDE as
$\dfrac{\partial u}{\partial \bar z} - iz \dfrac{\partial u}{\partial t} = \phi'(t) \tag{3}$
admits no local solutions near $0$ on $\Bbb R \times \Bbb C$ when $\phi$ is not analytic.  
