year 12 DE question

Hi, is the solution to a differential equation a function or a relation?
Thanks

An ordered pair is a set of inputs and outputs and represents a relationship between the two values. A relation is a set of inputs and outputs, and a function is a relation with one output for each input.

Please review:


If you still have questions, please come back and tell us your thoughts regarding the problem - and we will discuss further.
 
I disagree with Subhotosh a little on his definition of a function. I prefer to say that a function is a relation with no more than one output for each input.

Another words for a given input you may not have an output. For example, if f(x) = 1/(x-2), then f(2) has not output.
To defend Subhotosh (what am I thinking), he is correct if all inputs come from the domain of the relation.
 
I disagree with Subhotosh a little on his definition of a function. I prefer to say that a function is a relation with no more than one output for each input.

Another words for a given input you may not have an output. For example, if f(x) = 1/(x-2), then f(2) has not output.
To defend Subhotosh (what am I thinking), he is correct if all inputs come from the domain of the relation.
That's the difference between a mathematician and an engineer !! [DNE (does-not-exist) and infinity (exists and as many forms)]
 
Hi, is the solution to a differential equation a function or a relation?
Thanks
In general, the solution to a differential equation is a set or family of similar differentiable functions. Typically, you need additional information to reduce the answer to a specific differentiable function, which of course is one element in the set of all differentiable functions, which is one element in the set of all functions, which is one element in the set of all relations.

But I find it most helpful to think of the solution to a differential equation in terms of a family of functions having almost identical properties.

I would welcome the comments of Subhotosh Khan and Jomo on this answer.
 
In general, the solution to a differential equation is a set or family of similar differentiable functions. Typically, you need additional information to reduce the answer to a specific differentiable function, which of course is one element in the set of all differentiable functions, which is one element in the set of all functions, which is one element in the set of all relations.
But I find it most helpful to think of the solution to a differential equation in terms of a family of functions having almost identical properties.
I would welcome the comments of Subhotosh Khan and Jomo on this answer.
You said:

I find it most helpful to think of the solution to a differential equation in terms of a family of functions having almost identical properties.

I agree with this definition enthusiastically. This concept of family of solutions will be very useful while considering stability of solutions of DE.

- Comment from a now-Certified-Vagabond ex-engineer.​
 
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