# ODE initial values and continuity

• mathman44
In summary: It should address the uniqueness aspect of the question. In summary, the conversation discusses finding a continuous solution for t > 0 to the initial value problem y'(t)+p(t)y(t)=0, y(0)=1, where p(t)=2 for t [0,1] and p(t)=1 for t > 1. One proposed solution is y = e-2t for t [0,1], but the question remains about finding a continuous solution for t > 0. The mention of a theorem on existence and uniqueness of solutions may provide insight into this issue.
mathman44

## Homework Statement

Find a continuous y(t) for t > 0 to the initial value prob:

$$y'(t)+p(t)y(t)=0, y(0)=1$$
where
$$p(t)=2$$ for 0 < t < 1
$$p(t)=1$$ for t > 1

and determine if the soln is unique.

## The Attempt at a Solution

By standard ODE techniques I arrive at

$$y=\exp(-2t)$$ for 0 < t < 1
$$y=\exp(-t)$$ for t > 1

The problem is that this soln y(t) isn't continuous.. what's wrong here? As far as I know the only way to do this is to solve for y(t) in both intervals of t.

Last edited:

One problem with this is that p(t) isn't defined at 0, yet your initial condition is y(0). So your differential equation isn't defined at 0, but you are supposed to find a solution y(t) that is defined at 0.

Oops... I should have said that p(t) is 2 for t [0,1].

Then y = e-2t is a solution that is continuous on [0, 1], the interval that contains the initial value t = 0.

I think that's what we're looking for, but your text should have a theorem about existence and uniqueness of solutions of DEs. See what that theorem has to say about this situtation.

But the question is asking for a continuous soln for t > 0, not just t belonging to [0,1].

Take a look at the theorem I mentioned.

## 1. What is an initial value in ODE?

An initial value in ODE (Ordinary Differential Equations) is a starting point or condition for the solution of a differential equation. It is a known value of the dependent variable at a specific point in the independent variable's domain.

## 2. How is continuity defined in ODE?

In ODE, continuity refers to the smoothness of a function. It means that the function is defined and has no abrupt changes or discontinuities. In other words, the function is continuous if its value changes gradually as the independent variable changes.

## 3. What is the significance of initial values in solving ODEs?

The initial values are essential in solving ODEs as they provide a starting point for the solution. They help determine the unique solution to the differential equation and also ensure that the solution is continuous. Without initial values, the solution would not be well-defined.

## 4. How are initial values and continuity related in ODEs?

Initial values and continuity are closely related in ODEs. The initial values are used to determine the unique solution to the ODE, and this solution must be continuous. If the initial values are not chosen carefully, the resulting solution may not be continuous, and the solution will not be valid.

## 5. What are some methods for ensuring continuity in ODE solutions?

There are several methods for ensuring continuity in ODE solutions, such as using the Euler method or higher-order numerical methods, which involve approximating the solution at smaller intervals. Another approach is to use adaptive step sizes, where the step size is adjusted based on the smoothness of the solution, ensuring continuity. Additionally, using a more accurate initial value can also help ensure continuity in the solution.

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