Math Help - Differential Equation Curve

1. Differential Equation Curve

A curve rises from the origin continuously in the
xy plane into the first quadrant. The area under the curve from (0, 0) to (x, y) is 1/3 the area of the rectangle with these points as opposite vertices. Find the differential equation satisfied by the curve.

A curve rises from the origin continuously in the

xy plane into the first quadrant. The area under the curve from (0, 0) to (x, y) is 1/3 the area of the rectangle with these points as opposite vertices. Find the differential equation satisfied by the curve.

Let $y = f(x)$be the curve so the area under the curve is

$\int_0^x f(t)\, dt$

and the rectangle is $x f(x)$ so the equation to your question is

$
\int_0^x f(t)\, dt = \frac{1}{3} x f(x)
$

and differentiating gives

$
f(x) = \frac{1}{3} ( x f'(x) + f(x)).
$