ifr20 Later this semester, we will study the Heaviside "step" function H (t), which is defined to be 0ift<0 H(t) = if t >0 (See the image above for the graph of the Heaviside function.) Find all possible solutions (if any) to the initial value problem below, where H(t) is the Heaviside function: y (1) = H(t)y(t), with y(0) = yo- %3D In your work, carefully specify the number of solutions for each possible value of yo. Note: since H(1) is a discontinuous function, Theorems 2.4.1 and 2.4.2 in our textbook do not apply. So, in theory, anything could happen! Hints: since the values of H(t) are either 0 or 1, first find all possible solutions to y Oy and %3D separately to y = 1y. Then see if any two of those solutions can be "pieced together" to make a single solution y(t) to y = H(t) y with the given initial condition that y(0) = yo- Another Hint: a "solution" to an initial value problem must be differentiable on all of its domain.
ifr20 Later this semester, we will study the Heaviside "step" function H (t), which is defined to be 0ift<0 H(t) = if t >0 (See the image above for the graph of the Heaviside function.) Find all possible solutions (if any) to the initial value problem below, where H(t) is the Heaviside function: y (1) = H(t)y(t), with y(0) = yo- %3D In your work, carefully specify the number of solutions for each possible value of yo. Note: since H(1) is a discontinuous function, Theorems 2.4.1 and 2.4.2 in our textbook do not apply. So, in theory, anything could happen! Hints: since the values of H(t) are either 0 or 1, first find all possible solutions to y Oy and %3D separately to y = 1y. Then see if any two of those solutions can be "pieced together" to make a single solution y(t) to y = H(t) y with the given initial condition that y(0) = yo- Another Hint: a "solution" to an initial value problem must be differentiable on all of its domain.
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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Transcribed Image Text:Later this semester, we will study the Heaviside "step" function H(t), which is defined to be
S0 ift<0
1 ift >0
H(t) =
(See the image above for the graph of the Heaviside function.)
Find all possible solutions (if any) to the initial value problem below, where H(t) is the Heaviside
function:
y (t) = H(t)y(t), with y(0) = yo-
In your work, carefully specify the number of solutions for each possible value of yo.
Note: since H(t) is a discontinuous function, Theorems 2.4.1 and 2.4.2 in our textbook do not
apply. So, in theory, anything could happen!
Hints: since the values of H(t) are either 0 or 1, first find all possible solutions to y = 0y and
separately to y = 1y. Then see if any two of those solutions can be "pieced together" to make a
single solution y(t) to y = H(t) y with the given initial condition that y(0) = yo.
Another Hint: a "solution" to an initial value problem must be differentiable on all of its domain.
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