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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) =

Advanced Engineering Mathematics
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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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) =
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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