Graphs of solutions of the equation dy/dx = y^2. The graph y = Cx^4 for various

Question

Graphs of solutions of the equation dy/dx = y^2. The graph y = Cx^4 for various values of C. Show that y(x) = Cx^4 defines a one-parameter family of differentiable solutions of the differential equation xy' = 4y (Fig. 1.1.9). Show that y(x) = {-x^4 if x 0 defines a differentiable solution of xy' = 4y for all x, but is not of the form y(x) = Cx^4. 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.

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