0008.tex

\begin{exerciseStatement}


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



 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 \(t= -4 \). 

\[ {y'} = \cos\left({y} + t\right) \hspace{2em}
          y( -2 )= 2 \]
\end{exerciseStatement}
    
\begin{exerciseAnswer} 
\[y_p( -4 )\approx 2.0 \]
\end{exerciseAnswer}