0002.tex

\begin{exerciseStatement}


 Use \verb|https://sagecell.sagemath.org/| to run the SageMath code \verb|t,y = var('t y'); plot_slope_field(t*y/9+t/3, (t,-5,5), (y,-5,5))| producing the direction field for the ODE \( {y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \). 



 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 \(t= 2 \). 

\[ {y'} = \frac{1}{9} \, t {y} + \frac{1}{3} \, t \hspace{2em}
          y( 0 )= -2 \]
\end{exerciseStatement}
    
\begin{exerciseAnswer} 
\[y_p( 2 )\approx -1.8 \]
\end{exerciseAnswer}