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(a) Show that y(x) = Cx4 defines a one-parameter family of
Chapter 1, Problem 48P(choose chapter or problem)
Problem 48P
(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.
Questions & Answers
QUESTION:
Problem 48P
(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.
ANSWER:
Solution
Step 1 of 4
In this problem, we need to prove the given conditions.