Consider the function v(x, t) that satisfies the PDE vx + 8xvt=0 for x>0 and t> 0, and the initial condition v(x,0) = 0. (a) Apply the Laplace transform in t to the PDE and derive an expression for V/V, where V(x, s) = L(v(x, t)) is the Laplace transform in t of v. VT = (b) Integrate to find V in the form V(x, s) = C(s)g(x, s), where C(s) comes from the constant of integration and g(0,s) = 1. g(x, s) = AP (c) If u satisfies the boundary condition v(0, t) = 6t then find C(s). C(s) = P (d) If v(x, t) = f(t - A)u(t - A), where u is the unit step function, then find A(x) and f(t). A(x) = & P f(t) =
Consider the function v(x, t) that satisfies the PDE vx + 8xvt=0 for x>0 and t> 0, and the initial condition v(x,0) = 0. (a) Apply the Laplace transform in t to the PDE and derive an expression for V/V, where V(x, s) = L(v(x, t)) is the Laplace transform in t of v. VT = (b) Integrate to find V in the form V(x, s) = C(s)g(x, s), where C(s) comes from the constant of integration and g(0,s) = 1. g(x, s) = AP (c) If u satisfies the boundary condition v(0, t) = 6t then find C(s). C(s) = P (d) If v(x, t) = f(t - A)u(t - A), where u is the unit step function, then find A(x) and f(t). A(x) = & P f(t) =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 77E
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Question
![Consider the function v(x, t) that satisfies the PDE
vx + 8xvt 0 for x>0 and t > 0,
and the initial condition v(x, 0) = 0.
(a) Apply the Laplace transform in t to the PDE and derive an expression for Vx/V, where V(x, s) = L(v(x, t)) is the Laplace transform in t of v.
VT
=
V
(b) Integrate to find V in the form V(x, s) = C(s)g(x, s), where C(s) comes from the constant of integration and g(0, s) = 1.
g(x, s)
=
GP
(c) If u satisfies the boundary condition v(0, t) = 6t then find C(s).
C(s) =
a P
(d) If v(x, t) = f(t - A)u(t - A), where u is the unit step function, then find A(x) and f(t).
A(x) =
P
f(t)
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ab74d7a-2aa2-41fb-ad77-66c3f7c23f53%2F15769b7e-f38c-4939-927e-a5157671fdd7%2Fx9tduui_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the function v(x, t) that satisfies the PDE
vx + 8xvt 0 for x>0 and t > 0,
and the initial condition v(x, 0) = 0.
(a) Apply the Laplace transform in t to the PDE and derive an expression for Vx/V, where V(x, s) = L(v(x, t)) is the Laplace transform in t of v.
VT
=
V
(b) Integrate to find V in the form V(x, s) = C(s)g(x, s), where C(s) comes from the constant of integration and g(0, s) = 1.
g(x, s)
=
GP
(c) If u satisfies the boundary condition v(0, t) = 6t then find C(s).
C(s) =
a P
(d) If v(x, t) = f(t - A)u(t - A), where u is the unit step function, then find A(x) and f(t).
A(x) =
P
f(t)
=
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