
Diffy Q
A function y = g(x) is described by some geometric property of its graph. Write a differential equation of the form dy/dx = f(x, y) having the function g as its solution (or as one of its solutions).
"The line tangent to the graph of g at the point (x, y) intersects the xaxis at the point (x/2, 0)."

Hi
The equation of the line tangent to the graph of g at the point whose abscissa is $\displaystyle x_0$ is $\displaystyle y = g'(x_0)(xx_0)+g(x_0)$
This line intersects the xaxis at the abscissa given by the equation $\displaystyle 0 = g'(x_0)(xx_0)+g(x_0)$ or $\displaystyle x = x_0  \frac{g(x_0)}{g'(x_0)}$
We know that this abscissa is $\displaystyle \frac{x_0}{2}$ therefore for every $\displaystyle x_0$ inside g domain
$\displaystyle \frac{x_0}{2} = x_0  \frac{g(x_0)}{g'(x_0)}$
$\displaystyle g'(x_0) = \frac{2\: g(x_0)}{x_0}$
g satisfies the differential equation $\displaystyle y' = \frac{2\: y}{x}$
