MAT223H1 Study Guide - Final Guide: Ordinary Differential Equation, Partial Differential Equation, Generalized Eigenvector

My teacher is not very helpful, so please write cleanly and explain your steps so I can learn.

Please answer all 5 parts and draw a square around each answer.

Using a technique of integration and the separation-of-variables technique, you'll discover a function whose antiderivative models the shape of an ideal cable suspended from its endpoints. The function depends on a parameter b, which is determined by the properties of the cable, such as the density of the material out of which it is made.

Using principles from physics, it can be shown that when a perfectly flexible, uniformly dense, and inextensible cable is suspended from its endpoints, then it will assume a shape of a curve y = f(x) satisfying the following second-order differential equation:

where ρ is the density of the material of the cable, g is the acceleration due to gravity, and T is the magnitude of the tension in the cable at its lowest point.

We assume our cable is as in the diagram above and y = f(x) is its shape function; thus, not only does y satisfy the "hanging cable differential equation" above, we also have y '(0) = 0. Hence, making the substitution u =dy/dx in our the hanging cable equation, we obtain the first order, initial-value problem satisfied by u:

where b = ρg/T Solve the preceding initial-value problem, entering your (explicit) solution in the form u = g(x) where g is a function of x in which the constant b is present.