Final answer:
This question involves solving an associated homogeneous equation and defining functions y and w in terms of u(x) and u'(x). Define (y) and (w) using u(x), solving y''y' = 1.
Step-by-step explanation:
The given question involves the concept of solving an associated homogeneous equation. We are given the function y1(x) as a solution of the homogeneous equation y"y' = 1 and y1 = 1. We are also asked to define the functions y and w in terms of u(x) and u'(x).
To solve this, we let y = u(x)y1 and w(x) = u'(x). By substituting these expressions into the given equation, we can obtain a new equation that involves only the unknown function u(x). By solving this new equation, we can find the function y(x).
In solving the associated homogeneous equation y''y' = 1, where y1(x) is a known solution with y1 = 1, we introduce new functions (y) and (w) in terms of u(x) and u'(x). Let y = u(x)y1 and w(x) = u'(x). Substituting these expressions into the given equation yields a new equation involving only the unknown function u(x). By solving this equation, we can determine u(x) and subsequently find the function y(x). This method allows us to leverage the given solution y1(x) to express the solution of the original differential equation in terms of an arbitrary function u(x) and its derivative.