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Differential Equations

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Properties of the System of Equations - 1

DIFFEQ-NEDJ1Q

Consider the system of linear differential equations $X'=AX$ where $A$ is a $2$ by $2$ real matrix. If $A$ has two distinct real eigenvalues $\lambda_{1}$ and $\lambda_{2}$, $V_{1}$ and $V_{2}$ are two corresponding eigenvectors to $\lambda_{1}$ and $\lambda_{2}$ respectively, then which of the following MUST be true?

I. The general solution to the system is given by $X=c_{1}V_{1}e^{\lambda_{1}t}+c_{2}V_{2}e^{\lambda_{2}t}$ where $c_{1}$ and $c_{2}$ are real numbers.

II. The Wronskian of $X_{1}=V_{1}e^{\lambda_{1}t}$ and $X_{2}=V_{2}e^{\lambda_{2}t}$ is not 0.

III. The fundamental matrix $\left(\begin{array}{cc} V_{1}e^{\lambda_{1}t} & V_{2}e^{\lambda_{2}t} \end{array}\right)$ is invertible.

A

I only

B

II only

C

I and III only

D

II and III only

E

All of the three