How to solve set of differential equations?

Question

I have these sets of equations,

$x'_1(t)=x_2(t)$

$x'_2(t)=-x_2(t)-\lambda _2(t)$

$\lambda '_1(t)=-x_1(t)$

$\lambda '_2(t)=-\lambda _1(t)+\lambda _2(t)$

where $x^T(0)=\left[2\quad 5\right]$ and $\lambda _i^T(t=tf)=0$ where $tf=5$

I know this is simply a form of;

$x'(t)=Ax$

Edit: Added matrix A

where $A=\left[ \begin{array}\\ 0&1&0&0\\ 0&-1&0&-1\\ -1&0&0&0\\ 0&0&-1&1 \end{array}\right]$

But couldn't solve.

Answer 1

The solution to a matrix differential equation $x'(t)=Ax(t)$ is known to be

$$x(t)=e^{tA}x_0,$$

where $e^{X}$ denotes the matrix exponential and $x_0$ is the initial value.