# How to prove: Uniqueness of solution to first order autonomous ODE

• Jösus
In summary, the conversation discusses a proof for the statement that the initial value problem for a function f(x) has a unique solution on any interval where it can be defined. The person asking for help is looking for a direct proof without using complicated tools from the theory of ODE's. They suggest assuming the existence of two solutions, g and h, and proving that they cannot both exist. However, it is pointed out that the equations should actually be dg/dt = f(g(t)) and dh/dt = f(h(t)), making it unclear why the derivatives should be equal. It is then suggested that if f(x_0) ≠ 0, there is a unique solution on an interval containing x_0, but when
Jösus
Hello!

I would like to prove the following statement: Assume $f\in C^{1}(\mathbb{R})$. Then the initial value problem $\dot{x} = f(x),\quad x(0) = x_{0}$ has a unique solution, on any interval on which a solution may be defined.

I haven't been able to come up with a proof myself, but would really like to see a direct proof, not using too serious tools from a sophisticated theory of ODE's. I would very much appreciate it if someone could help me out.

Hello Jösus!
Jösus said:
… the initial value problem $\dot{x} = f(x),\quad x(0) = x_{0}$ has a unique solution, on any interval on which a solution may be defined.

So you can assume that there is a solution g, and you have to prove that there can't be two solutions, g and h.

So suppose dg/dx = dh/dx = f.

Then … ?

Correct me if I'm wrong, but shouldn't the equation read $dg/dt = f(g(t)), \quad dh/dt = f(h(t))$, and thus there would be no apparent reason for these derivatives to be equal?

I have thought about it some more, and found that if $f(x_{0}) \neq 0$ then there is an interval containing $x_{0}$ on which a unique solution exists (the equation is separable, so a simple integration trick works). When f reaches a zero in finite time, so that the inteval on which this solution is defined is cut off there may be extensions to the solutions beyond the problematic points. If, say, $f(a) = 0$ then setting $x(t) = a$ for t larger than (or smaller than if a is on the right of our starting point) will, I believe, do the trick. It is solutions of this type that aren't always unique, but with the requirement $f \in C^{1}(\mathbb{R})$ it should work. Any new ideas?

## 1. How can I prove that a first order autonomous ODE has a unique solution?

One way to prove uniqueness of solution for a first order autonomous ODE is by using the Picard-Lindelöf theorem. This theorem states that if the function in the ODE is Lipschitz continuous with respect to the dependent variable, then the solution is unique.

## 2. Can I use any other methods besides the Picard-Lindelöf theorem to prove uniqueness of solution?

Yes, there are other methods that can be used to prove uniqueness of solution for a first order autonomous ODE. These include the method of separation of variables, the method of integrating factors, and the method of undetermined coefficients.

## 3. Is the uniqueness of solution to a first order autonomous ODE always guaranteed?

No, the uniqueness of solution is not always guaranteed. In some cases, the ODE may have multiple solutions or no solutions at all. It is important to check for any conditions or constraints on the problem that may affect the uniqueness of the solution.

## 4. How does the initial value affect the uniqueness of solution for a first order autonomous ODE?

The initial value is a crucial factor in determining the uniqueness of solution for a first order autonomous ODE. In order for the solution to be unique, the initial value must be within the domain of the function and satisfy any given constraints or conditions.

## 5. Can I use numerical methods to prove uniqueness of solution for a first order autonomous ODE?

No, numerical methods cannot be used to prove uniqueness of solution. These methods can only approximate the solution and do not provide a rigorous proof of uniqueness. Analytical methods must be used to prove the uniqueness of solution for a first order autonomous ODE.

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