# Interval of Existance - Diff. Eq.

• theuniverse
In summary, the text says that there exists a unique solution to the initial value problem that satisfies the differential equation in some interval. The solution is a differentiable function and it satisfies the differential equation in some interval which is contained in the interval a < x < b.
theuniverse

## Homework Statement

Without solving the DE, determine an interval in which a solution to the given initial value problem is certain to exist. Is it certain to be a unique solution?
[(pi^2/16) - t^2]y' + y^(1/2)tan(t) = 0
Initial Value: y(3pi/8) = 1

## Homework Equations

dy/dt + p(x)y = q(x)

## The Attempt at a Solution

- It says not to solve it so I'm thinking of analyzing the functions and their intervals. I know that sqrt(y) is t>0 and y>0 and I also know that tan(t) is -pi/2<t<pi/2.
- I am not sure how the expression of the derivative is to be used.
- How is it all connecting eventually to the initial value I am given.

That's all I can think of, so it would be very helpful if you can provide me some guidelines how to continue from here.
Thanks so much!

Last edited:
If you put your equation in standard form, it becomes
$$y' = \frac{-tan(t)\cdot y^{1/2}}{\frac{\pi ^2}{16} - t^2}$$

That denominator is 0 when t = +/- pi/4.

so -pi/4 < t < pi/4 as well as anything small than -pi/4 or bigger than pi/4 is the interval in which all my solutions will have to be correct? and both -pi/4 and pi/4 will be the vertical asymptotes. Now how do I relate the initial value: y(3pi/8) = 1 into it?

I think that where this problem is going is that your interval is -pi/4 < t < pi/4. There's also the problem of tan(t) being undefined at odd multiples of pi/2.

In the DE text I have at hand, the existence and uniqueness theorem has this to say:
"Let f(x, y) and $\partial f /\partial y$ be continuous functions of x and y in some rectangle R of the xy plane defined by a < x < b, c < y < d. If (x0, y0) is any point of R, then there exists a unique solution of the initial-value problem
y'(x) = f(x, y(x)), y(x0) = y0.

This solution is a differentiable function and satisfies the differential equation in some interval x0 - h < x < x0 + h which is contained in the interval a < x < b."
Linear Algebra and Differential Equations, 2nd Ed. Charles G. Cullen.

## 1. What is an interval of existence?

An interval of existence refers to the range of values for which a solution to a differential equation is valid. It indicates the domain in which the solution makes sense and is applicable.

## 2. How is an interval of existence determined?

An interval of existence is determined by examining the initial conditions of the differential equation and solving for the range of values that satisfy those conditions. It can also be determined by analyzing the behavior of the equation and identifying any potential singularities or discontinuities.

## 3. What happens if a solution falls outside the interval of existence?

If a solution falls outside the interval of existence, it is considered invalid and cannot be used to accurately represent the behavior of the system described by the differential equation. It may also indicate that the initial conditions were not properly defined or that the equation itself is not a suitable model for the system.

## 4. Can an interval of existence be infinite?

Yes, an interval of existence can be infinite if the solution to the differential equation remains valid for an unbounded range of values. This can occur in cases where the equation has no singularities or discontinuities and the solution remains well-behaved for all values of the independent variable.

## 5. Are there any techniques for extending the interval of existence?

Yes, there are several techniques for extending the interval of existence, such as applying transformations or substitutions to the equation, using different methods of solving the equation, or incorporating boundary conditions. However, these techniques may not always be successful and may require additional considerations and assumptions.

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