Let us solve the following equation via separation of variables8²u(x, t)Ət²8²u(x, t)dx²a)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)0a³X(z)da²√C X(x) 0 = 14T(t) 0a²T(t)3²02²0₂²8²T(t)= 4CT(t)0²b)We are interested in the oscillatory solution given byX(x) = a₁ cos (√√|C|z) + a2 sin (√√C|x)If the boundary conditions areu(x = 0,t) = 0u(x = 1,t) = 0Identify 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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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)