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Convert $\frac{d^{2} x}{d t^{2}}+x=0$ to a first-order differential equation. Solve over the interval $[0, \pi]$ with $h=\frac{\pi}{10}$ assuming the initial conditions $x(0)=1$ and $x^{\prime}(0)=0$. Use the program linearode.
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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.
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