Question

Determine the largest interval (a,b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the initial value problem below.

xy′′′−7y'+eˣy = x⁷−4,y(6)=1,y′(6)=0,y′′(6)=2

Answer 1

Without the specific details of Theorem 1, it is not possible to determine the exact interval (a, b). The IVP's conditions and unique theorem must be analyzed to establish the interval where the theorem guarantees a unique solution.

Step-by-step explanation:

The student is asking about a theorem that guarantees the existence and uniqueness of solutions to an initial value problem (IVP) involving a third-order ordinary differential equation.

Determining the largest interval (a, b) for which such a theorem applies requires analyzing the IVP and theorem in question, typically a version of the Existence and Uniqueness Theorem which might require the function and its derivatives to be continuous or to satisfy certain conditions within that interval.

However, without the specific details of Theorem 1 that is being referred to, it's not possible to provide an exact interval.

The IVP provided includes the third derivative term xy''', a first derivative term -7y', an exponential function term e^x y, and the non-homogeneous part x^7 - 4. Along with the initial conditions at x = 6, these give us starting points for the solution but are not enough to conclude the interval without the theorem's conditions. Therefore, in order to determine the interval, one would need to evaluate the theorem in context of the differential equation's coefficients and the initial conditions.