Consider the equationdydx = y 4xx y . (i)(a) Show that Eq. (i) can be rewritten asdydx =

Chapter 2, Problem 30

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Consider the equationdydx = y 4xx y . (i)(a) Show that Eq. (i) can be rewritten asdydx = (y/x) 41 (y/x); (ii)thus Eq. (i) is homogeneous.(b) Introduce a new dependent variable v so that v = y/x, or y = xv(x). Express dy/dx interms of x, v, and dv/dx.(c) Replace y and dy/dx in Eq. (ii) by the expressions from part (b) that involve v anddv/dx. Show that the resulting differential equation isv + xdvdx = v 41 v,orxdvdx = v2 41 v . (iii)Observe that Eq. (iii) is separable.(d) Solve Eq. (iii), obtaining v implicitly in terms of x.(e) Find the solution of Eq. (i) by replacing v by y/x in the solution in part (d).(f) Draw a direction field and some integral curves for Eq. (i). Recall that the right sideof Eq. (i) actually depends only on the ratio y/x. This means that integral curves havethe same slope at all points on any given straight line through the origin, although theslope changes from one line to another. Therefore, the direction field and the integralcurves are symmetric with respect to the origin. Is this symmetry property evident fromyour plot?