Differential equations problem

[andmcg]

Homework Statement

A curve with Cartesian equation y=f(x) passes through the origin. Lines drawn parallel to the coordiante axes through and arbitrary poing of the curve form a rectangle with two sides on the axes. The curve divides every such rectangle into two regions A and B, one of which has and area equal to n times the other. Find all such functions f.

Homework Equations

The Attempt at a Solution

Obviously f(x)=cx where c is a real number works, but there must be others. Any ideas?

 

[Eynstone]

$$\int$$ydx (from 0 to x) = kxy ( for k = +- 1/n+1 or n/n+1).
This gives y = k(x y' + y) ; hence y =const. x^(1-k)

 

[HallsofIvy]

The area below the curve is, of course, $\int_0^{x} f(t)dt$ while the area above the curve is $\int_0^{x} f(x)- f(t) dt= xf(x)- \int_0^{x} f(t) dt$ Saying one is n times the other means that $n\int_0^{x}f(t)dt= xf(x)- \int_0^x f(t)dt$ so $xf(x)= (n+1)\int_0^x f(t) dt$.

Differentiating both sides of that with respect to x will give you a differential equation for f(x)- xdf/dx+ f(x)= (n+1)f(x) so that xdf/dx= n f(x). That is separable and easily integrable.