valid.ts where thealways be transformed into separable equations by a change of the dependent variable. Problem 25 illustrates how to solve first-order homogeneous equations.25. N Consider the equationdydx=y-4xx-yalid.s where thea. Show that equation (29) can be rewritten asdydx=(yx)-41-(yx)thus equation (29) is homogeneous.b. Introduce a new dependent variable v so that v=yx, or y=xv(x). Express dydx in terms of x,v, and dvdx.c. Replace y and dydx in equation (30) by the expressions from part b that involve v and dvdx. Show that the resulting differential equation isvxdvdx=v-41-vorxdvdx=v2-41-vObserve that equation (31) is separable.d. Solve equation (31), obtaining v implicitly in terms of x.e. Find the solution of equation (29) by replacing v by yx in the solution in part d.f. Draw a direction field and some integral curves for equation (29). Recall that the right-hand side of equation (29) actually depends only on the ratio yx. This means that integral curves have the same slope at all points on any given straight line through the origin, although the slope changes from one line to another. Therefore, the direction field and the integral curves are symmetric with respect to the origin. Is this symmetry property evident from your plot?The method outlined in Problem 25 can be used for any homogeneous equation. That is, the substitution y=xu(x) transforms a homogeneous equation into a separable equation. The latter equation can be solved by direct integration, and then replacing v by yx gives the solution to the original equation. In each of Problems 26 through 31:athematical contexts. The homogeneousus equations that will occur in Chapter 3