Question

Question: Solve the system

$\frac{\mathbf{d}\overrightarrow{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{A}\overrightarrow{\mathbf{x}}$

for a 2x2 matrix A with complex eigenvalues$\mathit{p}\mathbf{\pm }\mathit{i}\mathit{q}$ .

1. Find${\mathit{e}}^{\mathbf{2}\mathbf{\pi }\mathbf{i}}$ .

Short answer

Answer:

The solution is 1

Step-by-step solution

Euler’s formula

$e^{it}=cost+isint$

Explanation of the solution 

To find the value of $e^{i2\pi }$.

By using the Euler’s formula as follows.

$$\begin{gathered} e^{it}=cost+isint \\ {e}^{i2\pi }=cos\left(2\pi \right)+isin\left(2\pi \right) \\ {e}^{i2\pi }=1+0 \\ {e}^{i2\pi }=1 \end{gathered}$$

Hence, the value of $e^{i2\pi }$ is 1.