11. Consider the differential equation dy dt (a) Show that the constant function yı(t) = 0 is a solution. (b) Show that there are infinitely many other functions that satisfy the differen- tial equation, that agree with this solution when t ≤ 0, but that are nonzero when t > 0. [Hint: You need to define these functions using language like "y(t) = ... when t ≤0 and y(t) = ... when t > 0."] (c) Why doesn't this example contradict the Uniqueness Theorem?

11. Consider the differential equation dy dt (a) Show that the constant function yı(t) = 0 is a solution. (b) Show that there are infinitely many other functions that satisfy the differen- tial equation, that agree with this solution when t ≤ 0, but that are nonzero when t > 0. [Hint: You need to define these functions using language like "y(t) = ... when t ≤0 and y(t) = ... when t > 0."] (c) Why doesn't this example contradict the Uniqueness Theorem?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

I have provided the answer. Part (b), I see how the textbook got the solutions but do not understand how the t values determine the piecewise function.

11. Consider the differential equation
dy
dt
(a) Show that the constant function yı(t) = 0 is a solution.
(b) Show that there are infinitely many other functions that satisfy the differen-
tial equation, that agree with this solution when t ≤0, but that are nonzero
when t > 0. [Hint: You need to define these functions using language like
"y(t) = ... when t ≤0 and y(t) = ... when t > 0."]
(c) Why doesn't this example contradict the Uniqueness Theorem?

11. (a) If y₁ (t) = 0, then dy₁/dt = 0 = y₁/t².
(b) For any real number c, let
Yc (t) =
{
for t ≤ 0;
ce-¹/1, fort > 0.
0,
The function ye(t) satisfies the equation for all
t = 0. It is 0 for t < 0 and nonzero for t > 0.

Definition Definition Group of one or more functions defined at different and non-overlapping domains. The rule of a piecewise function is different for different pieces or portions of the domain.

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