What about a question where the matrix formed is A= 1 1 upper row and 1 4 lower row and the conditions are u1(x,0)=sin x and u2(x,0)=x² solving the boundary value problem using the matrix method

In this problem, you are asked to solve a boundary value problem using the **matrix method**, with an initial matrix \( A \) and boundary conditions given as \( u_1(x, 0) = \sin(x) \) and \( u_2(x, 0) = x^2 \).

The **matrix** you mentioned is:

\[
A = \begin{pmatrix}
1 & 1 \\
1 & 4
\end{pmatrix}
\]

### Step-by-step approach to solving using the matrix method:

#### 1. Define the general problem:
Suppose the system of equations is represented as:

\[
\frac{d\mathbf{u}}{dt} = A \mathbf{u}(x,t)
\]

Where \( \mathbf{u}(x,t) = \begin{pmatrix} u_1(x,t) \\ u_2(x,t) \end{pmatrix} \), and the initial conditions are given by:

\[
u_1(x, 0) = \sin(x) \quad \text{and} \quad u_2(x, 0) = x^2
\]

#### 2. Eigenvalues and Eigenvectors of matrix \( A \):
To solve this system, we begin by finding the **eigenvalues** and **eigenvectors** of matrix \( A \). The matrix differential equation will be easier to solve in terms of the eigenvalues and eigenvectors.

1. **Find the eigenvalues** of \( A \):

\[
\text{Determinant}(A - \lambda I) = 0
\]

\[
\begin{vmatrix}
1 - \lambda & 1 \\
1 & 4 - \lambda
\end{vmatrix} = 0
\]

Expanding this determinant:

\[
(1 - \lambda)(4 - \lambda) - 1 = 0
\]

\[
\lambda^2 - 5\lambda + 3 = 0
\]

The eigenvalues \( \lambda_1 \) and \( \lambda_2 \) are the roots of this quadratic equation:

\[
\lambda = \frac{5 \pm \sqrt{25 - 12}}{2} = \frac{5 \pm \sqrt{13}}{2}
\]

So, the eigenvalues are:

\[
\lambda_1 = \frac{5 + \sqrt{13}}{2}, \quad \lambda_2 = \frac{5 - \sqrt{13}}{2}
\]

2. **Find the eigenvectors** corresponding to each eigenvalue.**

For \( \lambda_1 \):

Solve \( (A - \lambda_1 I)\mathbf{v} = 0 \).

For \( \lambda_2 \):

Solve \( (A - \lambda_2 I)\mathbf{v} = 0 \).

#### 3. Diagonalize the system:
Using the eigenvalues and eigenvectors, we can diagonalize the matrix \( A \). This will simplify the differential equation into a set of decoupled equations, which are easier to solve.

#### 4. Solve the decoupled system:
For each eigenvalue \( \lambda_i \), solve the decoupled differential equation:

\[
\frac{d\mathbf{v}_i}{dt} = \lambda_i \mathbf{v}_i
\]

The solution for each component will be of the form:

\[
\mathbf{v}_i(t) = \mathbf{v}_i(0) e^{\lambda_i t}
\]

#### 5. Apply initial conditions:
Using the given initial conditions \( u_1(x, 0) = \sin(x) \) and \( u_2(x, 0) = x^2 \), you can determine the constants involved in the solution. This will give the specific solution to the boundary value problem.

#### 6. Combine the solutions:
Finally, express the general solution \( \mathbf{u}(x,t) \) in terms of the original basis (using the eigenvectors and the solutions from the decoupled system) to find \( u_1(x,t) \) and \( u_2(x,t) \).

---

### Would you like further details on any specific step?

Here are some follow-up questions related to this process:

1. How do we find eigenvectors from eigenvalues in matrix equations?
2. What is the purpose of diagonalizing a matrix in differential equations?
3. Can you explain the decoupling of systems in more detail?
4. How do boundary conditions affect the final solution?
5. How is matrix exponentiation applied in solving systems of differential equations?

#### Tip: Always ensure that the matrix is diagonalizable when using the eigenvalue method, as it simplifies solving differential equations effectively.

Solve the boundary value problem using matrix methods for A = [1 1; 1 4] with initial conditions u1(x,0)=sin(x) and u2(x,0)=x². This problem involves linear algebra, eigenvalues, eigenvectors, and matrix diagonalization. Suitable for advanced undergraduate students.

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