Let denote the solution to the initial value problem (a) Show that (...

Let denote the solution to the initial value problem

(a) Show that

(b) Argue that the graph of is decreasing for x near zero and that as x increases from zero, decreases until it crosses the line y = x, where its derivative is zero.

(c) Let x* be the abscissa of the point where the solution curve crosses the line . Consider the sign of and argue that has a relative minimum at x*.

(d) What can you say about the graph of for x > x*?

(e) Verify that y = x - 1 is a solution to dy/dx = x - y and explain why the graph of always stays above the line y = x - 1.

(f ) Sketch the direction field for dy/dx = x - y by using the method of isoclines or a computer software package.

(g) Sketch the solution using the direction field in part (f) .