Why the domain of a DE solution must be an interval?

[cathode-ray]

Hi everyone!

I started studying differential equations, but I still didn't understand why the domain of the solutions must be defined in an interval. For example the solution of the following initial value problem:

$$\frac{dy}{dt} + \frac{1}{t}y=0,y(1)=1$$

is given by y(t)=1/t that has it's domain in the interval ]0,+infinity[. Why can't it be defined in the set ]-infinity,0[ U ]0,+infinity[?

It just makes sense to me if we are using the solution to model a physical situation but mathematically I don't get it.

 

[AlephZero]

Why can't it be defined in the set ]-infinity,0[ U ]0,+infinity[?

It can be defined on that set if you want to.

The general solution on ]-infinity,0[ U ]0,+infinity[ is

y = A/t when t > 0
y = B/t when t < 0
where A and B are two independent constants.

The boundary conditions at a point can only affect the solution in a connected region containing that point. Your boundary condtion at t =1 fixes the value of A, but tells you nothing about the value of B.

So there is not usually much added value in lumping disconnected regions together into one "general solution".

 

[cathode-ray]

The boundary conditions at a point can only affect the solution in a connected region containing that point. Your boundary condtion at t =1 fixes the value of A, but tells you nothing about the value of B.

So there is not usually much added value in lumping disconnected regions together into one "general solution".

I hadn't thought about this. Now it makes sense.

Thanks!