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Differential Equations

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Properties of the Eigenvalue Method

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Which of the following is always true about a system of linear differential equations $X'=AX$ where $A$ is an $n\times n$ matrix?

I) If $\lambda$ is an eigenvalue and $v$ is a corresponding eigenvector, then $X=ve^{\lambda t}$ is a solution to the system.

II) If $X_{1},X_{2},\cdots, X_{n}$ are linearly independent solutions to the system, then the general solution must be in the form $X=c_{1}X_{1}+c_{2}X_{2}+\cdots+c_{n}X_{n}$ for any constants $c_{1},c_{2},\cdots,c_{n}$.

III) If $X_{1},X_{2},\cdots,X_{n}$ are solutions to the system, then the Wroskian of $X_{1},X_{2},\cdots,X_{n}$ is not zero.

A

I only

B

I and II only

C

I and III only

D

II and III only

E

I, II and III