# (a) Verify that y = -1 / (x + c) is a one-parameter family of solutions of the differential equation...

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^{2}.

(b) Since f (x, y) = y^{2} and Ï‘f/Ï‘y = 2y 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' = y^{2}, y(0) = 0.

**Theorem 1.2.1**

Let R be a rectangular region in the xy-plane defined by a ‰¤ x ‰¤ b, c ‰¤ y ‰¤ d that contains the point (x_{0}, y_{0}) in its interior. If f (x, y) and Ï‘f /Ï‘y are continuous on R, then there exists some interval I_{0}: (x_{0} h, x_{0} + h), h > 0, contained in [a, b], and a unique function y(x), defined on I0, that is a solution of the initial value problem (2).

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**Related Book For**

## A First Course in Differential Equations with Modeling Applications

**ISBN:** 978-1305965720

11th edition

**Authors:** Dennis G. Zill

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