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a) After using the ansatz u(x, t) = X(z)T(t), which equations do you get? Pick only two. (below C' denotes a constant) a²T(t) a²X(z) a³X(z) √3CT(t) 0 a³T(t) 2² = 1 T(t) √C X(1) 0 a³X(z) 8₂² = C X(T) Əz² a³r(t) 2² 3CT(t) = = — X(z)

a) After using the ansatz u(x, t) = X(z)T(t), which equations do you get? Pick only two. (below C' denotes a constant) a²T(t) a²X(z) a³X(z) √3CT(t) 0 a³T(t) 2² = 1 T(t) √C X(1) 0 a³X(z) 8₂² = C X(T) Əz² a³r(t) 2² 3CT(t) = = — X(z)

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter13: Conic Sections
Section13.1: Circles
Problem 48PS
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c) Given the solution for X(x), what is the solution for T(t)? o T(t) = e 3n²x²t ○ T(t) = b₁ cos(3²¹nt) + b2 sin(3²¹_¹nt) O T(t) = b₁ cos(3nnt) + b₂ sin(3nnt) O T(t) = b₁ cos(√3nnt) + b₂ sin(√3nπt) d) [D/HD] A different PDE has a full general solution is of the form 3n²²t u(x,t) = Σ sin(n × ππ)xe n=1 which of the following is a possible solution consistent with the initial conditions u(x, t0) = = sin(87x) + sin(157x) O u(2,t)= sin(15 × #T) xe 3x225m²t Ⓒu(x, t) = sin(8 × πx) × e+³×64²t + sin(15 × T) × e+3x225x²t Ou(x, t) sin(8 × π) × е -3x64m²t Ⓒu(x, t) = sin(8 × π™) × е 3x225m²t + sin(15 xx) x e -3x64x²t ○ u(x, t) = sin(8 × π) × (cos(√3×8 × t) +5 sin(√3 x 8 x πt))
Transcribed Image Text:c) Given the solution for X(x), what is the solution for T(t)? o T(t) = e 3n²x²t ○ T(t) = b₁ cos(3²¹nt) + b2 sin(3²¹_¹nt) O T(t) = b₁ cos(3nnt) + b₂ sin(3nnt) O T(t) = b₁ cos(√3nnt) + b₂ sin(√3nπt) d) [D/HD] A different PDE has a full general solution is of the form 3n²²t u(x,t) = Σ sin(n × ππ)xe n=1 which of the following is a possible solution consistent with the initial conditions u(x, t0) = = sin(87x) + sin(157x) O u(2,t)= sin(15 × #T) xe 3x225m²t Ⓒu(x, t) = sin(8 × πx) × e+³×64²t + sin(15 × T) × e+3x225x²t Ou(x, t) sin(8 × π) × е -3x64m²t Ⓒu(x, t) = sin(8 × π™) × е 3x225m²t + sin(15 xx) x e -3x64x²t ○ u(x, t) = sin(8 × π) × (cos(√3×8 × t) +5 sin(√3 x 8 x πt))
Let us solve the following equation via separation of variables J²u(x, t) Ət² = 3. a²u(x, t) ?х2 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) Ə²X(1) J²X(z) a²T(t) 842 √3CT(t) a²T(t) 84² = 1 T(t) 8²X(z) Əx² √C X(T) = C X(T) 8x² Ər² a²T(t) 3C T(t) b) We are interested in the oscillatory solution given by X(x) = a₁ cos ( $(√|C|x) + 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 sin(n+x) O X(x) = a₂ cos(NTT) O X(z) = a2 sin((2n-1)) = = = X(T) = OX(x) = a₂ cos((2n-1) -x)
Transcribed Image Text:Let us solve the following equation via separation of variables J²u(x, t) Ət² = 3. a²u(x, t) ?х2 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) Ə²X(1) J²X(z) a²T(t) 842 √3CT(t) a²T(t) 84² = 1 T(t) 8²X(z) Əx² √C X(T) = C X(T) 8x² Ər² a²T(t) 3C T(t) b) We are interested in the oscillatory solution given by X(x) = a₁ cos ( $(√|C|x) + 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 sin(n+x) O X(x) = a₂ cos(NTT) O X(z) = a2 sin((2n-1)) = = = X(T) = OX(x) = a₂ cos((2n-1) -x)
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