To convert a second-order differential equation to a system of first-order differential equations, we introduce a new variable for each derivative of the function, up to one less than the order of the equation.

For the given second-order differential equation, we can introduce two new variables:

Let \(y_1=x\) and \(y_2=x^{\prime}\) (the derivative of \(x\) with respect to \(t\) ).

Then, the original second-order differential equation can be written as two first-order differential equations:

1. \(y_1^{\prime}=y_2\)

2. \(y_2^{\prime}=-y_1\)

With the initial conditions \(y_1(0)=1\) and \(y_2(0)=0\).

Now, we can solve this system of first-order differential equations using the program 'linearode '. However, as an Al model, I don't have the capability to directly run programs or software. I can provide a general outline of how you might use a numerical method like Euler's method or Runge-Kutta method to solve this system, but you'll need to implement it in your programming environment.

Here's a general outline using Euler's method:

1. Initialize \(y_1\) and \(y_2\) with their initial conditions.

2. For each step from \(t=0\) to \(t=\pi\) with step size \(h=\pi / 10\) :

- Calculate the new \(y_1\) and \(y_2\) using the equations \(y_1^{\prime}=y_2\) and \(y_2^{\prime}=-y_1\).

- Update \(y_1\) and \(y_2\) with these new values.

3. The solution to the differential equation is the final values of \(y_1\) and \(y_2\) at \(t=\pi\).

Remember, Euler's method is a simple numerical method and may not provide an accurate solution for larger intervals or more complex differential equations. For more accuracy, consider using a more sophisticated method like the Runge-Kutta method.