Existence of unique solutions to a first order ODE on this interval

In summary, the conversation discusses different ways to prove that a first order ode has a unique solution on the interval (1,infinity). One method is to show that the derivative and the partial derivative of the function are continuous. However, this only proves existence on an interval within (1,infinity). Another method is to ensure that the function is bounded, which guarantees that all solutions are defined on the entire interval. The conversation also mentions a theorem that states that if a solution to an IVP is defined on a certain interval and cannot be extended further, then a limit exists at the end point of the interval and the solution cannot be extended beyond that point.
  • #1
thidmir
5
1
TL;DR Summary
I am trying to find if there is a way to prove the existence and uniqueness of a solution
to a first order ODE on an interval including infinity.
I am trying to find a way to prove that a certain first order ode has a unique
solution on the interval (1,infinity). Usually the way to do this is to show that
if x' = f(t,x) (derivative with respect to t), then f(t,x) and the partial derivative with respect to f are continuous.
However, this would show that a solution exists only on an interval inside (1,infinity).
Is there any way to show that a solution exists on the entire interval?
 
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  • #2
In general a solution is not obliged to be defined for all ##t\ge 0##. For example $$\dot x=x^2,\quad x(0)=1$$
If in equation ##\dot x=f(t,x)## the function ##f## is such that $$|f(t,x)|\le c_1+c_2|x|,\quad (t,x)\in\mathbb{R}_+\times\mathbb{R}^m$$ then all the solutions to such a system are defined in ##[0,\infty)##. There are a lot of other different sufficient conditions for that
 
  • #3
The following theorem is also useful.
Assume that $$f(t,x)\in C^1((t_1,t_2)\times D,\mathbb{R}^m)$$ where ##D\subset\mathbb{R}^m## is an open domain.

Assume also that $$|f(t,x)|\le c$$ for all ##(t,x)\in (t_1,t_2)\times D##.

Theorem. Let ##x(t)## be a solution to the following IVP
$$\dot x=f(t,x),\quad x(t_0)=x_0\in D,\quad t_0\in(t_1,t_2).$$ Assume that ##x(t)## is defined in ##[t_0,t^*),\quad t^*<t_2 ## and can not be extended longer than ##t^*##. Then the following limit exists
$$\lim_{t\to t^*-}x(t)=x^*$$ and $$x^*\notin D.$$
 
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Related to Existence of unique solutions to a first order ODE on this interval

1. What is a first order ODE?

A first order ordinary differential equation (ODE) is a mathematical equation that describes the relationship between a single variable and its derivative (with respect to that variable). It is typically written in the form of dy/dx = f(x), where y is the dependent variable, x is the independent variable, and f(x) is a function of x.

2. What is a unique solution to a first order ODE?

A unique solution to a first order ODE is a solution that satisfies the equation and any initial conditions given. This means that there is only one possible function that can satisfy the equation and initial conditions, and any other solution would not be considered a valid solution.

3. What does it mean for a solution to be unique on an interval?

A solution is considered unique on an interval if it is the only solution that satisfies the equation and initial conditions within that interval. This means that there are no other functions that can satisfy the equation and initial conditions within that specific interval.

4. How do you determine if a first order ODE has a unique solution on a given interval?

To determine if a first order ODE has a unique solution on a given interval, you can use the existence and uniqueness theorem. This theorem states that if the function f(x) in the equation dy/dx = f(x) is continuous on the given interval and satisfies Lipschitz's condition, then there exists a unique solution to the ODE on that interval.

5. What is Lipschitz's condition?

Lipschitz's condition is a mathematical concept that ensures the uniqueness of a solution to a first order ODE. It states that the function f(x) must satisfy the inequality |f(x) - f(y)| ≤ L|x - y| for all values of x and y in the given interval, where L is a constant. This condition guarantees that the function does not change too rapidly, allowing for a unique solution to exist.

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