Frankly, I'm not sure what some of these problems have to do with Calculus, but since they were assigned by a Calculus instructor, I'll assume they do.

The first one doesn't seem so bad at first.

For what values of c does the equation $\displaystyle \ln{x}=cx^2$ have exactly one solution?

I tried graphing to compare the two functions, and all I've been able to ascertain from the graphs is that c must be less than one. I don't know any way to solve this equation analytically. Ideas?

The other one, oh boy.

When a flexible cable of uniform density is suspended between two fixed points and hangs of its own weight, the shape $\displaystyle y=f(x)$ of the cable must satisfy a differential equation of the form $\displaystyle \frac{d^2y}{dx^2}=k\sqrt{1 + \frac{dy}{dx}^2}$ where k is a positive constant. Let $\displaystyle z=\frac{dy}{dx}$ in the differential equation. Solve the resulting first-order differential equation (in z) and then integrate to find y.

Maybe it's because I've been ill this weekend, but I'm completely stumped. I don't even know where to start.

Thanks for the help!