Solve the following system of differential equations:

 with , Where 

To solve the given system of differential equations \(\frac{dy(t)}{dt} = Ay(t)\), where \(A\) is the given matrix and \(y(0)\) is the initial condition vector, we can use the matrix exponential method.

The solution for \(y(t)\) is given by:

\[ y(t) = e^{At} \cdot y(0) \]

First, we need to find the matrix exponential \(e^{At}\). The matrix exponential can be calculated using the Taylor series expansion:

\[ e^{At} = \sum_{k=0}^{\infty} \frac{(At)^k}{k!} \]

For our problem, we have \(A\) as:

\[ A = \begin{bmatrix} 2 & -5 & -11 \\ 0 & -2 & -9 \\ 0 & 1 & 4 \end{bmatrix} \]

Let's calculate \(e^{At}\):

\[ e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \ldots \]

Now, we calculate each term of the series:

1. \( A^0 = I \)
2. \( A^1 = A \)
3. \( A^2 = A \cdot A \)
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