(a) Show that y(x) = Cx 4 defines a one-parameter family of differentiable solutions of the differential...
Question:
(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.
Step by Step Answer:
Differential Equations And Linear Algebra
ISBN: 9780134497181
4th Edition
Authors: C. Edwards, David Penney, David Calvis