Please solve & show steps... ### Determining Intervals of Existence for Initial Value Problems**Problem Statement:**Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist.**Equation:**\[ t(t - 18)y' + y = 0, \quad y(8) = 1 \]**Interval Specification:**\[ \boxed{\ } < t < \boxed{\ } \]**Explanation:**The given initial value problem involves solving a differential equation of the form \(t(t - 18)y' + y = 0\) with an initial condition \(y(8) = 1\). To determine the interval of t in which a solution is guaranteed to exist, it is essential to identify the values of t where the coefficient of \(y'\) (i.e., \(t(t - 18)\)) might cause singularities or discontinuities. In this equation, singularities occur where \(t(t - 18) = 0\), that is, at \(t = 0\) and \(t = 18\). The solution is therefore guaranteed to exist in an interval that does not include these points of singularity. Given the initial condition \(y(8) = 1\), we look for an interval around t = 8 which does not include 0 or 18. Thus, the interval in which the solution is certain to exist will be:\[ 0 < t < 18 \]In this interval, the differential equation remains well-behaved and does not hit any points of singularity.

Name: Please solve & show steps... ### Determining Intervals of Existence fo
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: Please solve & show steps... ### Determining Intervals of Existence for Initial Value Problems**Problem Statement:**Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist.**Equ