Question

Find all curves y=y(x) such that the tangent to the curve at any point (xo, y(xo)) intersects the x axis at xI=(xo)^3.

Answer 1

@myininaya

Answer 2

The tangent line at any point \((x_0,y_0)\) to the curve \(y=y(x)\) has the general form
\[y-y_0=y'(x_0)(x-x_0)\]
You want to guarantee that the curves pass through the point \((x_i,0)=({x_0}^3,0)\), so plugging in \(x={x_0}^3\) and \(y=0\) into the tangent line equation gives the ODE,
\[0-y_0=y'(x_0)({x_0}^3-x_0)\]
which is linear in \(y_0=y(x_0)\), and also separable. Solve whichever way you like.

Answer 3

Thanks!