(a) Verify that y=tan(x + c) is a one-parameter family of solutions of the differential equation y' = 1+ y². O Differentiating y = tan(x + c) we get y'=csc(x + c) or y' = 1 + y². O Differentiating y = tan(x + c) we get y'= tan²(x + c) or y' = 1 + y². O Differentiating y=tan(x + c) we get y' = 1 + tan²(x + c) or y' = 1 + y². Differentiating y =tan(x + c) we get y' = 1 + sec²(x + c) or y' = 1 + y². O Differentiating y =tan(x + c) we get y' = sec(x + c) or y' = 1 + y².
(a) Verify that y=tan(x + c) is a one-parameter family of solutions of the differential equation y' = 1+ y². O Differentiating y = tan(x + c) we get y'=csc(x + c) or y' = 1 + y². O Differentiating y = tan(x + c) we get y'= tan²(x + c) or y' = 1 + y². O Differentiating y=tan(x + c) we get y' = 1 + tan²(x + c) or y' = 1 + y². Differentiating y =tan(x + c) we get y' = 1 + sec²(x + c) or y' = 1 + y². O Differentiating y =tan(x + c) we get y' = sec(x + c) or y' = 1 + y².
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:(a) Verify that y = tan(x + c) is a one-parameter family of solutions of the differential equation y' = 1 + y².
O Differentiating y =tan(x + c) we get y' = csc(x + c) or y' = 1 + y².
O Differentiating y = tan(x + c) we get y' = tan²(x + c) or y' = 1 + y².
O Differentiating y = tan(x + c) we get y' = 1 + tan²(x + c) or y' = 1 + y².
O Differentiating y = tan(x + c) we get y' = 1 + sec²(x + c) or y' = 1 + y².
O Differentiating y = tan(x + c) we get y' = sec(x + c) or y' = 1 + y².
(b) Since f(x, y) = 1 + y² and af/ay = 2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initial-value problem y'= 1 + y², y(0) = 0.
y =
Even though x = 0 is in the interval (-2, 2), explain why the solution is not defined on this interval.
Since tan(x) is discontinuous at x = t
the solution is not defined on (-2, 2).
(c) Determine the largest interval I of definition for the solution of the initial-value problem in part (b). (Enter your answer using interval notation.)
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