Final answer:
The question asks to solve a differential equation using separation of variables and integration, which is a college-level mathematics topic. An initial condition is required to find a particular solution; otherwise, only a general solution can be derived.
Step-by-step explanation:
The student's question relates to finding the particular solution to the differential equation dy/dx = (x-2) e⁻²y. To solve this differential equation, one would typically use separation of variables by isolating all terms with y on one side of the equation and all terms with x on the other side. After separating, we would integrate both sides of the equation with respect to their respective variables. This process would lead to a general solution, which could then be used to find a particular solution if an initial condition is given. Since no initial condition is provided, we would only be able to derive the general solution. The solution to this problem involves both integration techniques and the understanding of exponential functions, which are usually covered in college-level calculus courses.
To clarify, the other content provided seems irrelevant to solving this equation and might be related to different topics such as kinematics in physics or other mathematical problems like quadratic equations, which are out of the context of this particular differential equation solution.