(a) Show that if is an integrating factor for an equation of the form
Consider the equation
in the domain . Using small line segments, sketch the flow directions in that domain. Show that is an integrating factor for the equation. Find the general solution of the equation, and sketch the family of solutions that occupies the larger domain .
(b) The following example illustrates that the concept of integrating factor extends to higher-order equations. Multiply the equation
by , and show that the result takes the form , for some function to be determined. Find a particular solution such that with finite at , and sketch its graph in .