Diffy Q

• January 30th 2009, 05:45 AM
bearej50
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 x-axis at the point (x/2, 0)."
• January 30th 2009, 07:24 AM
running-gag
Hi

The equation of the line tangent to the graph of g at the point whose abscissa is $x_0$ is $y = g'(x_0)(x-x_0)+g(x_0)$

This line intersects the x-axis at the abscissa given by the equation $0 = g'(x_0)(x-x_0)+g(x_0)$ or $x = x_0 - \frac{g(x_0)}{g'(x_0)}$

We know that this abscissa is $\frac{x_0}{2}$ therefore for every $x_0$ inside g domain

$\frac{x_0}{2} = x_0 - \frac{g(x_0)}{g'(x_0)}$

$g'(x_0) = \frac{2\: g(x_0)}{x_0}$

g satisfies the differential equation $y' = \frac{2\: y}{x}$
• January 30th 2009, 07:45 AM
bearej50
thank you much