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 (Type your answer in interval notation.)

Answer 1

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

Step-by-step explanation:

The given initial value problem is:

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

To find the largest interval (a,b) for which Theorem 1 guarantees the existence of a unique solution, we need to analyze the given equation. The existence and uniqueness theorem guarantees a unique solution on an interval if the coefficients of the highest-order derivatives are continuous on that interval.

In this case, the coefficient of the highest-order derivative is exy. The exponential function is continuous for all values of x, so there are no restrictions on the interval (a,b). Therefore, the largest interval is (-∞,∞) in interval notation.