Let us solve the following equation via separation of variables 8²u(x, t) Ət² 8²u(x, t) dx² a) After using the ansatz u(x, t) = X(x)T(t), which equations do you get? Pick only two. (below C denotes a constant) 3³ X(z) CX(x) 3²X(x) 0 8²T(1) 2² √ACT(t) 0 a³X(z) da² √C X(x) 0 =1&T(t) a²T(t) 3² dz² 0₂² 8²T(t) 0² = 4CT(t) b) We are interested in the oscillatory solution given by X(x) = a₁ cos (√√C|z) + a2 sin (√√C|x) If the boundary conditions are u(x = 0,t) = 0 u(x = 1,t) = 0 Identify the correct form of the solution: O X(a)= a₂ cos(n*x) O X(x) = a2 sin(n=x) O X(x)= a2 cos((2n-1)=x) = 4 = X(x) O X(x)= a₂ sin((2n-1)=x)
Let us solve the following equation via separation of variables 8²u(x, t) Ət² 8²u(x, t) dx² a) After using the ansatz u(x, t) = X(x)T(t), which equations do you get? Pick only two. (below C denotes a constant) 3³ X(z) CX(x) 3²X(x) 0 8²T(1) 2² √ACT(t) 0 a³X(z) da² √C X(x) 0 =1&T(t) a²T(t) 3² dz² 0₂² 8²T(t) 0² = 4CT(t) b) We are interested in the oscillatory solution given by X(x) = a₁ cos (√√C|z) + a2 sin (√√C|x) If the boundary conditions are u(x = 0,t) = 0 u(x = 1,t) = 0 Identify the correct form of the solution: O X(a)= a₂ cos(n*x) O X(x) = a2 sin(n=x) O X(x)= a2 cos((2n-1)=x) = 4 = X(x) O X(x)= a₂ sin((2n-1)=x)
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:Let us solve the following equation via separation of variables
8²u(x, t)
Ət²
8²u(x, t)
dx²
a)
After using the ansatz u(x, t) = X(x)T(t), which equations do you get? Pick only two. (below C denotes a constant)
8²T(t)
3³ X(z) = C X(x)
3²X(z)
√4CT(t)
0
a³X(z)
da²
√C X(x) 0 = 14T(t) 0
a²T(t)
3²
02²
0₂²
8²T(t)
= 4CT(t)
0²
b)
We are interested in the oscillatory solution given by
X(x) = a₁ cos (√√|C|z) + a2 sin (√√C|x)
If the boundary conditions are
u(x = 0,t) = 0
u(x = 1,t) = 0
Identify the correct form of the solution:
O X(x)= a₂ cos(n*x) O X(x) = a2 sin(n=x)
O X(x)= a2 cos((2n-1)x)
=
= 4
= X(x)
O X(x)= a₂ sin((2n-1)=x)
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