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2 years ago in Applied Physics , Mathematics By Ankit

Does a function f exist that satisfies f’(x) = -a + b/f(x) with f(0) = 0, and is it unique?

This ODE arose in my modeling work on a physical process with a source and decay. The initial condition f(0)=0 creates a singularity because it places b/f(x) in the equation at x=0. Standard existence and uniqueness theorems seem to break down here. I'm trying to understand if a meaningful solution can even be defined, or if the problem needs to be reformulated to be well-posed mathematically.

DifferentialGeometry Ordinary Differential Equations

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By Shilpa A Answered 2 years ago

This is a classic singular initial value problem. The immediate issue is that at x=0, the equation is undefined due to division by f(0)=0. In my work with similar models, existence isn't automatically ruled out, but uniqueness almost always fails. You might find a solution by treating it as a separable equation and integrating, but the initial condition would be applied as a limit. I would recommend reformulating the problem physically; perhaps the condition is f(0)=ε, where ε is small, and then analyze the limit. Uniqueness is highly unlikely at the true singularity.

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Does a function f exist that satisfies f’(x) = -a + b/f(x) with f(0) = 0, and is it unique?

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