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1) Solving Partial Differential Equations Using Separation of Variables. Consider the following PDE: ди at ди дх +u with the initial condition u(x, 0) = 6e-3x Ju(x,0) a) Why is there only one initial condition, that is why do not need a statement about ? at b) Use separation of variables to find two ordinary differential equations, one for X(x) and one for T(t) where we assume u(x,t) = X(x)T(t). c) Solve each ordinary differential equation. (Hint: look at the form of the initial condition as a guide). d) Using the initial condition, find the solution u(x,t) that satisfies the PDE and the initial condition. e) Check to see that your solution works. д²и a²u 2) Let u be a solution of the equation: for 0≤x≤ and t≥ 0 with the boundary at² მx2. conditions u(0,t) = 0 and u(л,t) = 0 and the initial conditions u(x,0) = sin(x) - 2sin(3x) ди and at (x, 0) = 0. Find the general solution u(x,t) for t > 0.

1) Solving Partial Differential Equations Using Separation of Variables. Consider the following PDE: ди at ди дх +u with the initial condition u(x, 0) = 6e-3x Ju(x,0) a) Why is there only one initial condition, that is why do not need a statement about ? at b) Use separation of variables to find two ordinary differential equations, one for X(x) and one for T(t) where we assume u(x,t) = X(x)T(t). c) Solve each ordinary differential equation. (Hint: look at the form of the initial condition as a guide). d) Using the initial condition, find the solution u(x,t) that satisfies the PDE and the initial condition. e) Check to see that your solution works. д²и a²u 2) Let u be a solution of the equation: for 0≤x≤ and t≥ 0 with the boundary at² მx2. conditions u(0,t) = 0 and u(л,t) = 0 and the initial conditions u(x,0) = sin(x) - 2sin(3x) ди and at (x, 0) = 0. Find the general solution u(x,t) for t > 0.

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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1) Solving Partial Differential Equations Using Separation of Variables. Consider the following PDE: ди at ди дх +u with the initial condition u(x, 0) = 6e-3x Ju(x,0) a) Why is there only one initial condition, that is why do not need a statement about ? at b) Use separation of variables to find two ordinary differential equations, one for X(x) and one for T(t) where we assume u(x,t) = X(x)T(t). c) Solve each ordinary differential equation. (Hint: look at the form of the initial condition as a guide). d) Using the initial condition, find the solution u(x,t) that satisfies the PDE and the initial condition. e) Check to see that your solution works. д²и a²u 2) Let u be a solution of the equation: for 0≤x≤ and t≥ 0 with the boundary at² მx2. conditions u(0,t) = 0 and u(л,t) = 0 and the initial conditions u(x,0) = sin(x) - 2sin(3x) ди and at (x, 0) = 0. Find the general solution u(x,t) for t > 0.
Transcribed Image Text:1) Solving Partial Differential Equations Using Separation of Variables. Consider the following PDE: ди at ди дх +u with the initial condition u(x, 0) = 6e-3x Ju(x,0) a) Why is there only one initial condition, that is why do not need a statement about ? at b) Use separation of variables to find two ordinary differential equations, one for X(x) and one for T(t) where we assume u(x,t) = X(x)T(t). c) Solve each ordinary differential equation. (Hint: look at the form of the initial condition as a guide). d) Using the initial condition, find the solution u(x,t) that satisfies the PDE and the initial condition. e) Check to see that your solution works. д²и a²u 2) Let u be a solution of the equation: for 0≤x≤ and t≥ 0 with the boundary at² მx2. conditions u(0,t) = 0 and u(л,t) = 0 and the initial conditions u(x,0) = sin(x) - 2sin(3x) ди and at (x, 0) = 0. Find the general solution u(x,t) for t > 0.
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