Projects for chapter 4 nonlinear equations solvable by first-order techniques certain nonlinear second-order equations-namely, those with dependent or independent variables missing-can be solved by reducing them to a pair of first-order equations. this is accomplished by making the substitution w- dy/dx, where x is the independent variable. (a) to solve an equation of the form f(x, y') in which the dependent variable y is missing, setting wy' (so that w'-y) yields the pair of equations because fx, ) is a first-order equation, we have available the techniques of chapter 2 to solve it for w(x). once w(x) is determined, we integrate it to obtain y(x) using this method, solve x> 0 (b) to solve an equation of the form y"-f(y, y') in which the independent variable x is missing, setting dy/dx yields, via the chain rule, dw dw dy dx2 dx dy d dy f(y, y') is equivalent to the pair of equations dw dy thus, y" dy dx in equation (1) notice that y plays the role of the independent variable; hence, solving it yields w(y). then substituting w(y) into (2), we obtain a separable equation that determines y(x) using this method, solve the following equations: (c) suspended cable. in the study of a cable suspended between two fixed points (see figure 4.36 on page 234), one encounters the initial value problem dy dx y(o)-a y'(0)0 ci where a ( 0) is a constant. solve this initial value problem for y. the resulting curve is called a catenary 233