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. (x2 - 25)y'"' + exy = Inx, y| *, »{)=1,»[)=" (1) - (Type your answer in interval notation.)

Answer 1

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

Step-by-step explanation:

The initial value problem you've given is (x^2 - 25)y'' + exy = Inx, where y(*) = 1 and y'(*) = 0. We need to determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b). To do this, we can identify the interval where each coefficient is continuous and the right-hand side is continuous. Theorem 1 states that if the coefficients and the right-hand side are continuous within an interval (a, b), then there exists a unique solution on that interval.

In this case, the function Inx is continuous for x > 0, so let's focus on the other terms. Since x^2 - 25 is a polynomial, it is continuous for all real values of x. The exponential term exy is a composition of continuous functions, and it's continuous for all x and y. Therefore, the only interval we need to consider is the interval where the function y is defined, which is the entire real line (minus any possible singular points of the differential equation).

To summarize, Theorem 1 guarantees the existence of a unique solution on the interval (-∞, ∞).