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",-6y' + eˣy-x⁴-3, y(6)-1, y'(6)-0, y'',(6)= 2

Answer 1

The largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution to the initial value problem is (-∞, ∞).

The initial value problem is denoted as follows:

xy''' - 6y' + e^xy = x^4 - 3

y(6) = 1, y'(6) = 0, y''(6) = 2

The existence and uniqueness of a solution to y''' - 6y' + e^xy = x4 - 3 on an interval (a, b) are guaranteed by Theorem 1. To identify the greatest interval (a, b) where the theorem holds true, we must first evaluate the theorem's conditions and look for any limits on the equation.

Theorem 1 indicates that there exists a unique solution on any interval (a, b) because the presented equation is a linear homogeneous differential equation with constant coefficients (y''' - 6y' + e^xy = 0).

As a result, (-, ) is the greatest interval (a, b) where Theorem 1 guarantees the existence of a single solution to the initial value problem.