(a) Show that y(x) = Cx⁴ defines a one-parameter family of differentiable solutions of the differential equation xy' = 4y (Fig. 1.1.9).

(b) Show that

defines a differentiable solution of xy' = 4y for all x, but is not of the form y(x) = Cx⁴.

(c) Given any two real numbers a and b, explain why-in contrast to the situation in part (c) of Problem 47-there exist infinitely many differentiable solutions of xy' = 4y that all satisfy the condition y (a) = b.

FIGURE 1.1.9. The graph y = Cx⁴ for various values of C.

Part (c) of Problem 47

(c) Figure 1.1.8 shows typical graphs of solutions of the form y(x) = 1/(C - x). Does it appear that these solution curves fill the entire xy- plane? Can you conclude that, given any point (a, b) in the plane, the differential equation dy/dx = y² has exactly one solution y(x) satisfying the condition y(a) = b?

FIGURE 1.1.8. Graphs of solutions of the equation dy/dx = y².

y(x) = S-x4 [x4 if x < 0, if x 0