Exploring Interval of Definition for Differential Equation Solutions

[Ali Asadullah]

What is interval of definition of a solution of a Differential Equation?
How can we find the interval of definition of a differential equation?
What are the properties of this interval?
Is the solution of the DE and DE itself are continuous and differentiable on the interval?

 

[HallsofIvy]

The interval of definition of a solution to a differential equation is the largest interval upon which is it "sufficiently" differentiable (a solution of a second order differential equation must be twice differentiable, etc.) from which is follows that it is continuous, and on which it satisfies the given differential equation.

I'm not sure I have ever seen the phrase "interval of definition" applied to a differential equation itself before. But I will say that it doesn't make sense to talk about the equation itself being "continuous and differentiable". I assume you mean that the functions in the differential equation are differentiable. Again, "differentiable" implies "continuous" so it isn't necessary to say that.

 

[Ali Asadullah]

Dear Sir, if we are given an Initial value problem, say at x0, y=y0, then the interval of definition will be the largest interval containing the (x0,y0) on which the solution is "sufficiently differentiable. Am i right sir?