# Making up a differential equation with no real solutions

#### find_the_fun

##### Active member
Make up a differential equation that does not possess any real solutions.

step 1)consider the definition of solution
Any function $$\displaystyle \phi$$ defined on an interval I and possessing at least n derivatives that are continuous on I which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval

So it sounds to me like the task is to find a function that is not differentiable. Where I'm stuck is every function I know of is differentiable at some point and the question (I think) is asking for a DE that has no solutions over any interval.

#### MarkFL

Staff member
How about the square of a derivative being equated to a negative constant?

#### chisigma

##### Well-known member
... consider the definition of solution...

Any function $$\displaystyle \phi$$ defined on an interval I and possessing at least n derivatives that are continuous on I which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval
I'm of the opinion that the constraint of the continuity of the derivatives of the first n order may be removed and that allows the solution of a large number of pratical problems. For example let's consider the first order ODE...

$\displaystyle x^{\ '} = \mathcal {U} (t),\ x(0)=0\ (1)$

... where $\displaystyle \mathcal {U} (*)$ is the Heaviside Step Function...

Heaviside Step Function -- from Wolfram MathWorld

It is easy to verify that the solution of (1) is...

$\displaystyle x(t) = t\ \mathcal {U} (t)\ (2)$

Kind regards

$\chi$ $\sigma$

Last edited: