Final answer:
The differential equation's roots cannot be determined from the provided options, as they do not match the functional form of the solution derived from integrating the original equation.
Step-by-step explanation:
The student's question involves finding the roots of a differential equation, expressed as r' (the derivative of r with respect to x) plus a linear term in x. To solve this equation, we integrate both sides with respect to x. The integration of a constant yields a linear term, and the integration of a linear term yields a quadratic term. However, since we're integrating r', we end up with just r, and since the term -10x is linear, it integrates to a term of the form (-10/2)x^2 or -5x^2.
Ignoring the integration constant for now, the integrated equation on the right side becomes 137x - 5x^2. To solve for r, we take the derivative of both sides, giving us r' = 137 - 10x. This is actually just the original equation, which means that we've lost no information. Subsequently, if we consider constants, a general solution for r would be of the form r = C + 137x - 5x^2, where C is the integration constant.
None of the options provided, A through D, explicitly match the functional form of our derived solution, as they neither have the quadratic term nor an undetermined constant. This suggests that there might be a misunderstanding in the question leading to the conclusion that none of the given options accurately represents the roots of the differential equation provided in this context.