Final answer:
The student appears to be asking for help with an initial value problem that involves trigonometric functions and principles of physics, such as conservation of momentum or energy. However, due to missing information and potential typos in the provided question, a detailed step-by-step solution cannot be provided accurately.
Step-by-step explanation:
The question appears to involve solving an initial value problem involving trigonometric functions and potentially physics concepts, such as conservation of momentum or energy. The snippets reference solving an equation for cosine, relationships between intensities (I and I0), and the kinetic and potential energy of a system.
Unfortunately, there seems to be missing information or context that would allow for a step-by-step explanation of the solution process. Terms like '+= cos /' and '(\u03c0)=' are unclear, potentially due to typos, and without a clear equation, providing a comprehensive answer is not feasible.
From the given clues, such as initial kinetic energy and potential energy, it seems the solution involves the manipulation of equations in the context of physics to solve for a variable, and possibly using integral calculus for finding the average value over a complete cycle in a trigonometric function.
The complete question is: Solve the following initial value problem. dθdr=cos(−3πθ),r(0)=9 The solution is r(θ)= (Type an exact answer.)