# Showing uniqueness of complex ODE

In summary: I think what you're trying to say is that if f(z) is a function and z=1, then (1-z)*f'(z)=0.In summary, the student is trying to find a factor that makes the right side the derivative of a product, but is having difficulty doing so.

## Homework Statement

We are given (1-z)*f'(z)-3*f(z) = 0, f(0) = 2 valid on the open disk centered at 0 with radius 1 and told to prove there is a unique solution to the differential equation. The hint he gave was to find a factor that makes the left side the derivative of a product, but that didn't really help me.

## Homework Equations

The found factor was (1-z)^2, so you now have (1-z)^3*f'(z)-3(1-z)^2*f(z) = 0.

## The Attempt at a Solution

I integrated both sides as I thought you would do after making the LHS a derivative and you can solve it, but that doesn't tell me that it is unique or not... although it has been a long time since I've done existence/uniqueness so maybe that is all I have to do. I also tried looking at it as f'(z)=3/(1-z)*f(z) and I figured it had something to do with the analyticity of 3/(1-z), but I could never get anywhere with that idea.

It was just an extra credit problem on the last homework so we haven't gone into ODE's in the complex plane, but with Spring Break this week, won't get to see the solution for awhile so was just curious if anyone knew how to prove it, or could tell me if I was using the hint correctly.

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(1-z)*f'(z) - 3*f(z) = 0

(1-z)3*f'(z) - 3(1-z)2*f(z) = 0​

if the second equation is true, then either the first equation is true or z = 1

Well yes, that is true, but what I'm not able to figure out is how to show uniqueness, whether with the original equation or the new one.

well, the new equation is the same as

d/dt (1-z)3*f(z) = 0 …​

surely that means (1-z)3*f(z) is a constant, and sooo … ?

God... I feel stupid hahaha. I completely missed that when I wrote that down before.

## 1. How is the uniqueness of a complex ODE shown?

The uniqueness of a complex ODE can be shown by using the Picard-Lindelöf theorem, also known as the Cauchy-Lipschitz theorem. This theorem states that if the derivative of the function in the ODE is continuous and satisfies a Lipschitz condition, then the solution to the initial value problem is unique.

## 2. What is a Lipschitz condition?

A Lipschitz condition is a mathematical condition that ensures the derivative of a function does not change too quickly. This means that the rate of change of the function is bounded, which is necessary for the Picard-Lindelöf theorem to hold and for the uniqueness of a complex ODE to be shown.

## 3. Are there any other methods for showing uniqueness of a complex ODE?

Yes, there are other methods such as the method of characteristics and the method of energy estimates. However, the Picard-Lindelöf theorem is the most commonly used and reliable method for showing uniqueness of complex ODEs.

## 4. Can uniqueness of a complex ODE ever fail?

Yes, uniqueness of a complex ODE can fail if the function in the ODE does not satisfy the Lipschitz condition or if the initial conditions are not well-defined. In these cases, multiple solutions may exist for the same initial value problem.

## 5. How is the uniqueness of a complex ODE important in scientific research?

The uniqueness of a complex ODE is crucial in scientific research as it ensures that the solution to a problem is unique and can be relied upon. This is important in many fields such as physics, engineering, and economics where accurate and unique solutions are necessary for making predictions and understanding complex systems.

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