There is a system where Ax=B and A=[-1,2,3,6;0,1,-3,2;4,6,5,0;9,-2,-3,4] and x = [x1;x2;x3;x4] and B=[3;4;-2;-1]. There is also a given function: arcctg(sqrt(x))+(sqrt(4+x^2))^3 *y+2+sqrt(x)+cos(x)-(x^2 +1)*y = ln(1+sqrt(x)). Write in an open way y=y(x) and calculate the value of the function y(x4). Provide the answer with four decimal places.

We are dealing with two parts here:

1. **Solve the system of equations** \( A \cdot x = B \) to find \( x_1, x_2, x_3, x_4 \).
   
2. **Given the function**:
   \[
   \text{arcctg}(\sqrt{x}) + \left( \sqrt{4+x^2} \right)^3 \cdot y + 2 + \sqrt{x} + \cos(x) - (x^2 + 1) \cdot y = \ln(1 + \sqrt{x}),
   \]
   we need to express \( y \) as a function of \( x \) and compute \( y(x_4) \).

### Part 1: Solving the System of Equations \( A \cdot x = B \)

The matrix \( A \) and the vector \( B \) are given as:
\[
A = \begin{bmatrix}
-1 & 2 & 3 & 6 \\
0 & 1 & -3 & 2 \\
4 & 6 & 5 & 0 \\
9 & -2 & -3 & 4
\end{bmatrix},
\quad
B = \begin{bmatrix}
3 \\
4 \\
-2 \\
-1
\end{bmatrix}.
\]
We will solve this system of equations to find the values of \( x_1, x_2, x_3, x_4 \).

Let me calculate this first.The solution to the system \( A \cdot x = B \) is:
\[
x_1 \approx -0.4841, \quad x_2 \approx 0.6059, \quad x_3 \approx -0.7398, \quad x_4 \approx 0.5873.
\]

### Part 2: Express \( y = y(x) \)

We have the equation:
\[
\text{arcctg}(\sqrt{x}) + \left( \sqrt{4 + x^2} \right)^3 \cdot y + 2 + \sqrt{x} + \cos(x) - (x^2 + 1) \cdot y = \ln(1 + \sqrt{x}).
\]

Rearranging to solve for \( y \):
\[
y \left( \left( \sqrt{4 + x^2} \right)^3 - (x^2 + 1) \right) = \ln(1 + \sqrt{x}) - \text{arcctg}(\sqrt{x}) - 2 - \sqrt{x} - \cos(x).
\]
Thus,
\[
y = \frac{\ln(1 + \sqrt{x}) - \text{arcctg}(\sqrt{x}) - 2 - \sqrt{x} - \cos(x)}{\left( \sqrt{4 + x^2} \right)^3 - (x^2 + 1)}.
\]

### Part 3: Calculate \( y(x_4) \)

We now compute \( y \) when \( x = x_4 \approx 0.5873 \). Let me calculate this.It seems like I can't do more advanced data analysis right now. Please try again later.

If you have any questions or need clarification on the steps we've taken so far, feel free to ask! Here are some related questions for further exploration:

1. What is the general method for solving a system of linear equations like \( Ax = B \)?
2. How do we differentiate between arcctg and arctan functions?
3. What are some common applications of systems of equations in engineering or physics?
4. How does the derivative of the inverse trigonometric functions work?
5. Can we use numerical methods to approximate solutions for complex functions?

**Tip**: When solving complex functions, symbolic computation tools can be quite helpful to simplify expressions before numerical evaluations.

Learn how to solve a system of linear equations Ax = B where A is a 4x4 matrix and B is a vector, and compute the value of a complex function involving arcctg, logarithmic, and trigonometric expressions for y(x4). This problem covers linear algebra, advanced functions, and equation solving.

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