What is the interval of validity of a differential equation?

If y 0 > 0 − ∞ < t < 1 y 0 is the interval of validity . If y 0 = 0 − ∞ < t < ∞ is the interval of validity . If y 0 < 0 1 y 0 < t < ∞ is the interval of validity . On a side note, notice that the solution, in its final form, will also work if yo=0 y o = 0 .

On what interval is the solution of the IVP?

Since both p(t) and q(t) are discontinuous at x = ±2, the I.V.P. has a unique solution on the interval (−2,2).

What is interval of existence?

has a solution existing on some time interval containing t0 in its interior and that the solution is unique on that interval. Let’s say that an interval of existence is an interval containing t0 on which a solution of (1) exists. The following theorem indicates how large an interval of existence may be.

What do you know about existence and uniqueness of solutions of linear second order odes?

Uniqueness and Existence for Second Order Differential Equations. if p(t) and g(t) are continuous on [a,b], then there exists a unique solution on the interval [a,b]. We can ask the same questions of second order linear differential equations.

What is the interval of existence?

has a solution existing on some time interval containing t0 in its interior and that the solution is unique on that interval. Let’s say that an interval of existence is an interval containing t0 on which a solution of (1) exists.

What is uniqueness theorem in statistics?

A theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model).

What is existence theorem in differential equations?

Peano’s existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition. if y is absolutely continuous, y satisfies the differential equation almost everywhere and y satisfies the initial condition.

What is existence and uniqueness of solution?

The Existence and Uniqueness Theorem tells us that the integral curves of any differential equation satisfying the appropriate hypothesis, cannot cross. If the curves did cross, we could take the point of intersection as the initial value for the differential equation.

What is interval of validity?

What is the interval of validity? It is the largest interval of the x-axis which a) contains the initial condition, and b) on which the solution is defined and continuous.

What is existence and uniqueness theorem?

Existence and uniqueness theorem is the tool which makes it possible for us to conclude that there exists only one solution to a first order differential equation which satisfies a given initial condition.

What does the existence and uniqueness theorem tell us?

The Existence and Uniqueness Theorem tells us that the integral curves of any differential equation satisfying the appropriate hypothesis, cannot cross. If the curves did cross, we could take the point of intersection as the initial value for the differential equation.

Can a theorem be used to find an interval of validity?

Unlike the first theorem, this one cannot really be used to find an interval of validity. So, we will know that a unique solution exists if the conditions of the theorem are met, but we will actually need the solution in order to determine its interval of validity.

Are there intervals of validity for linear differential equations?

Intervals of validity for linear differential equations do not depend on the value of y o y o. Intervals of validity for non-linear differential can depend on the value of y o y o as we pointed out after the second theorem. So, let’s solve the IVP and get some intervals of validity.

Is the value of Yo y o the interval of validity?

The interval must contain to t o, but the value of yo y o, has no effect on the interval of validity. Let’s take a look at an example. First, in order to use the theorem to find the interval of validity we must write the differential equation in the proper form given in the theorem.