# Solving an integral equation with a simple trick

In a paper published in 2011 by Waldemar Klobus (Motion on a vertical loop with friction) an interesting integral equation (more precisely, a Volterra integral equation of the 2nd kind) appears:

which describes the motion of an object around a circular loop as shown in Fig.1 below:

and μ is the coefficient of friction, with respect to the tangential frictional force that the object experiences. V denotes the speed of the object normalized by a constant factor (which the author defines as the square root of *gR*, where *g* is the gravitational acceleration and *R* is the loop’s radius). Although the solution to Equation 1 is presented in the paper, the actual procedure for solving it is not. In this article, I will go through a step-by-step derivation of such a solution. By the way, V(0) is a constant which is derived in the paper based on some critical conditions, but in this article I will mainly focus on the general solution to Equation 1.

This integral equation can be solved by a simple trick, which is to use differentiation first. For example, if we take the 1st derivative of Eq. 1 we obtain

which is more easily written as:

Equation 2 is now a 1st-order differential equation for V² which is linear and has constant coefficients, therefore it has an exact…