Making up a differential equation with no real solutions

In summary, the conversation is about finding a differential equation that does not have any real solutions. The definition of a solution is discussed and it is suggested that the constraint of continuity of derivatives can be removed for practical purposes. An example of a first order ODE with a Heaviside Step Function is given to show that the solution can still be found.
  • #1
find_the_fun
148
0
Make up a differential equation that does not possesses 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 :confused:

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.
 
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  • #2
How about the square of a derivative being equated to a negative constant?
 
  • #3
find_the_fun said:
... 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 :confused:

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

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

Kind regards

$\chi$ $\sigma$
 
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FAQ: Making up a differential equation with no real solutions

What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is commonly used to model and describe natural phenomena in various fields of science.

How do you make up a differential equation?

To make up a differential equation, you need to specify the function and its derivatives, as well as any other relevant parameters. You can also use known relationships or laws from your field of study to create a differential equation.

What does it mean for a differential equation to have no real solutions?

If a differential equation has no real solutions, it means that there are no functions that satisfy the equation for all values of the independent variable. In other words, there are no real-valued functions that can be derived from the given equation.

Why is it important to study differential equations with no real solutions?

Studying differential equations with no real solutions can help us understand and describe complex systems and phenomena that cannot be explained by simple, real-valued functions. It also allows us to develop new mathematical tools and techniques for solving these types of equations.

What are some real-world applications of differential equations with no real solutions?

Differential equations with no real solutions are used to model and analyze a wide range of natural phenomena, including heat transfer, population dynamics, chemical reactions, and electrical circuits. They are also essential in fields such as engineering, physics, biology, and economics.

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