Mathematics: Find the general solution of x1? = x1 ? 2x2, x2? = 2x1 x2 using the eigenvalue method. do not use complex exponentials in your solution.

Find the general solution of x1? = x1 ? 2x2, x2?

Ответ:

leonidas117

09.01.2022

Guessing on how the system is expressed:

$\begin{bmatrix}x_1\\x_2\end{bmatrix}'=\begin{bmatrix}1&-2\\2&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$

The coefficient matrix has eigenvalues $\lambda=1+2i$ with corresponding eigenvectors $\begin{bmatrix}\pm i&1\end{bmatrix}^\top$ . This means the characteristic solution is

$\begin{bmatrix}x_1\\x_2\end{bmatrix}=Ce^{(1+2i)t}\begin{bmatrix}i\\1\end{bmatrix}$
$\begin{bmatrix}x_1\\x_2\end{bmatrix}=Ce^t(\cos2t+i\sin2t)\begin{bmatrix}i\\1\end{bmatrix}$
$\begin{bmatrix}x_1\\x_2\end{bmatrix}=C_1\begin{bmatrix}-e^t\sin2t\\e^t\cos2t\end{bmatrix}+C_2\begin{bmatrix}e^t\cos2t\\e^t\sin2t\end{bmatrix}$

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Ответ:

jeffylovesgreenbeans

28.08.2022

Mathematics

vertex form is f(x) = ( x-1)² + 3.

Step-by-step explanation:

Given : f(x) = 4 + x² – 2x.

To find : Which function in vertex form is equivalent to f(x) = 4 + x² – 2x.

Solution : We have given that f(x) = 4 + x² – 2x.

Rearrange in function : f(x) = x² – 2x + 4.

Vertex form of y = a(x-h)²+k .

Now, add and subtract (-1)² in f(x) .

f(x) = x² – 2x + 4 + (-1)²- (-1)².

By Squaring formula : (x +a)²= x²+ a²+2ax .

We can write x² – 2x + (-1)²- (-1)² +4 as ( x-1)²- (-1)² +4.

f(x) = ( x-1)²- (-1)² +4.

Now , completing square ,

f(x) = ( x-1)²- 1 +4.

f(x) = ( x-1)² + 3.

Therefore , vertex form is f(x) = ( x-1)² + 3.

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Find the general solution of x1? = x1 ? 2x2, x2? = 2x1 x2 using the eigenvalue method. do not use complex exponentials in your solution.

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