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Ordinary & Partial Differential Equations question bank (Differential Equations & Probability)

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Q1.

[Differential Equations & Probability Problem #1] Regarding eigenvalues of a real symmetric matrix A: Which of the following analytical formulations is correct?

(A)

Eigenvalues must always be positive integers

(B)

The matrix cannot be diagonalized under any orthogonal transformation

(C)

All eigenvalues are pure imaginary

(D)

All eigenvalues are strictly real, and eigenvectors from distinct eigenvalues are orthogonal

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Q2.

[Differential Equations & Probability Problem #3] Regarding Cauchy-Riemann equations for an analytic function f(z) = u(x,y) + i v(x,y): Which of the following analytical formulations is correct?

(A)

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

(B)

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

(C)

∂^2u/∂x^2 + ∂^2v/∂y^2 = 0 strictly

(D)

∂u/∂x + ∂v/∂y = 0 and ∂u/∂y + ∂v/∂x = 0

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