Solution Found!
(a) Verify that is a one-parameter family of solutions of
Chapter 1, Problem 31E(choose chapter or problem)
(a) Verify that y = 1/(x + c) is a one-parameter family of solutions of the differential equation \(y^{\prime}=y^{2}\).
(b) Since \(f(x, y)=y^{2}\) and \(\partial f / \partial y=2 y\) are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Find a solution from the family in part (a) that satisfies y(0) = 1. Then find a solution from the family in part (a) that satisfies y(0) = -1. Determine the largest interval I of definition for the solution of each initial-value problem.
(c) Determine the largest interval I of definition for the solution of the first-order initial-value problem \(y^{\prime}=y^{2}\), y(0) = 0. [Hint: The solution is not a member of the family of solutions in part (a).]
Text Transcription:
y^prime=y^2
f(x, y)=y^2
partial f / partial y=2y
y^prime=y^2
Questions & Answers
QUESTION:
(a) Verify that y = 1/(x + c) is a one-parameter family of solutions of the differential equation \(y^{\prime}=y^{2}\).
(b) Since \(f(x, y)=y^{2}\) and \(\partial f / \partial y=2 y\) are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Find a solution from the family in part (a) that satisfies y(0) = 1. Then find a solution from the family in part (a) that satisfies y(0) = -1. Determine the largest interval I of definition for the solution of each initial-value problem.
(c) Determine the largest interval I of definition for the solution of the first-order initial-value problem \(y^{\prime}=y^{2}\), y(0) = 0. [Hint: The solution is not a member of the family of solutions in part (a).]
Text Transcription:
y^prime=y^2
f(x, y)=y^2
partial f / partial y=2y
y^prime=y^2
ANSWER:Step 1 of 8
In this question, we have to verify that is a one-parameter family of solutions of the differential equation