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please solve using 10 decimal digits of precision 

The exact value of \( \alpha_1 \), calculated using the collocation method at the collocation point \( x = 2 \), is approximately:

\[
\alpha_1 = -0.5789473684
\]

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Here are 5 related questions for further exploration:
1. What is the general principle behind the collocation method in numerical analysis?
2. How does the residual function help in solving boundary value problems?
3. What would the solution look like for other collocation points, such as \( x = 1 \) or \( x = 3 \)?
4. How can the approximate solution \( u(x) \) be improved using additional terms?
5. What are some alternative methods to the collocation method for solving boundary value problems?

**Tip**: When applying the collocation method, the choice of collocation points can greatly influence the accuracy of the solution, so it's important to choose points wisely.

Use the collocation method with the collocation point x = 2 to find the value of α₁ in the equation u(x) = x - 1 + α₁(x² - 4x) with residual R(x) = 5 + 5x - x² + α₁(x² + 5x + 5).

Learn how to apply the collocation method to solve for α₁ in a boundary value problem using the collocation point x = 2. The method involves approximating the residual function and solving key equations in numerical analysis. Suitable for undergraduate-level students.

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