TITLE: Differential equation with no solutions
QUESTION [6 upvotes]: Is there an ordinary differential equation with no solutions? Precisely, I mean
1) not in the IVP context, just a simple ODE without IVP conditions,
2) not an ODE with complex solutions but no real ones (such as $(y') ^2+1=0$),
3) not of the form $y^{(n)}=y^{(n)}+c$.
Thanks

REPLY [8 votes]: Of course you can set up a differential equation $$y'=f(x,y)$$ with $f(x,y)=1$ if at least one of $x$, $y$ is irrational and $=0$ otherwise. Such a differential equation will have no solution, I guess. But as soon as there is a small disk with center $(x_0,y_0)$ on which $f$ is continuous Peano's existence theorem guarantees a solution $x\mapsto y(x)$ on some small $x$-interval $[x_0-h,x_0+h]$ satisfying the initial condition $y(x_0)=y_0$.