0000.tex

\begin{exerciseStatement}


 Use \verb|https://sagecell.sagemath.org/| to run the SageMath code \verb|t,y = var('t y'); plot_slope_field(sin(y/2), (t,-5,5), (y,-5,5))| producing the direction field for the ODE \( {y'} = \sin\left(\frac{1}{2} \, {y}\right) \). 



 Let \(y_p\) be the solution to the following IVP. Explain how to use its direction field to approximate the value of \(y_p\) at \(t= -3 \). 

\[ {y'} = \sin\left(\frac{1}{2} \, {y}\right) \hspace{2em}
          y( -1 )= -2 \]
\end{exerciseStatement}
    
\begin{exerciseAnswer} 
\[y_p( -3 )\approx -2.0 \]
\end{exerciseAnswer}