An ordinary Differential Equation which has no solution?

Question

Can we think of an Ordinary Differential Equation which has no solution?

How can we think of this ODE?

Like what $f$ should I use such that $\frac{dy}{dx} = f(x,y)$ with $y(x_{0}) = f(x_{0},y_{0})$.

From existence theorem I think I have to use a $f$ which is not Lipschitz continuous?

Answer 1

The Lipschitz condition is needed for uniqueness, not existence. As long as $f(x,y)$ is continuous in a neighbourhood of $(x_0, y_0)$ there is a solution.

For a simple example of non-existence, consider the d.e. $$ \dfrac{dy}{dx} = \cases{1 & if $xy < 0$\cr -1 & if $xy \ge 0$\cr} $$ with initial condition $y(0)=0$.

Answer 2

Derivatives have the Darboux property, that is, if $g:I\to\mathbb{R}$ is differentiable in the interval $I$ and $g'$ takes two distinct values, then it takes all the values in between. So in order to have an ODE $y'=f(x,y)$ with no solutions, we can take $f(x,y)=g(x)$ where $g$ is any function with jump discontinuities like the Dirichlet Function.