Recognizing Linear Homogeneous DEs from Direction Fields From your conclusions in Problem,...

Recognizing Linear Homogeneous DEs from Direction Fields From your conclusions in Problem, decide which of the direction fields in Fig. 4 represent linear homogeneous equations. Explain your reasons in each case.

Figure 4 Direction fields for Problem.

Problem

Direction Field Detective The direction fields of three DEs are shown in Fig. 3.

(a) Label the equations linear or nonlinear. Of the two that are linear, which is homogeneous and which is nonhomogeneous?

(b) Explain (algebraically) why the sura of two solutions of a homogeneous linear DE is a solution. Verify that the negative of such a solution is also a solution, so y = 0 must be a solution.

(c) Interpret the observations of part (b) for the direction fields in Fig. 3. Visualize two solutions and then their sura. Does the sura follow the direction field?

Figure 3 Direction fields for Problem.