Question

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.

Answer 1

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.