Final answer:
To find the differential equation for a function y=g(x), where the tangent at any point (x, y) passes through (-y, x), we determine that dy/dx = -1. This represents a constant negative rate of change of y with respect to x.
Step-by-step explanation:
The question involves finding a differential equation for which a function y=g(x) is a solution, given the geometric property that the tangent line to the graph of g at any point (x, y) passes through the point (-y, x). To solve this, we must write a differential equation dy/dx = f(x, y) that reflects this property.
Firstly, we know that the slope of the tangent line at any point is given by dy/dx. The slope is also described by the difference in y over the difference in x between two points on the line. In this case, the two points are (x, y) and (-y, x). Hence, the slope m is ((y - (-y)) / (x - x)), which simplifies to 2y / 0. However, division by zero is undefined, which suggests that our initial approach needs adjustment. Instead, recognize that a line passing through (x, y) and (-y, x) is perpendicular to the line y = x since their slopes are negative reciprocals (the slope of y = x is 1, so the slope of the perpendicular line is -1).
Therefore, we deduce that dy/dx must be equal to -1 for our differential equation, giving us the equation dy/dx = -1. This equation suggests that g(x) is a solution where the rate of change of y with respect to x is constant and negative, meaning that for an increase in x, there is an equal decrease in y.