Example

Use hw3plot.m to create a slopefield, numerical plot, and plot of the symbolic solution to the ODE $x' \,=\, 2\,t\,x$.

1. Separable equation
   $$\begin{eqnarray*} \frac{1}{x} \,dx &=& 2t \,dx \; \Longrightarrow \;\int \frac{1... ...} \quad\mbox{(for $C \,=\, \pm C_2 \,=\, \pm \mbox{e}^{C_1}$)} \end{eqnarray*}$$
   
   
   
   
   

2. ftx = `2*t*x';
   initax = [1,4,-2,0];
   t0 = 0; x0 = 0; C = 1;
   ftxsln = `C*exp(t$\wedge$2)';
3. slopef(ftx,N) results in
   
   
   
   
   
   Change initax = [-2,2,-2,2]; then slopef(ftx,N) results in
   
   
   
   
4. Running sol1 and placing pointer at about (1,1) results in
   
   
   
   
5. From the display we have t0 = 1.0932 and x0 = 0.95208 which we enter into init.m.
6. Using $x \,=\, C \mbox{e}^{t^2}$, we solve $x_0 \,=\, C \mbox{e}^{t_0^2}$; i.e.,
   $$\begin{eqnarray*} t_0 &=& 1.0932, \; x_0 \;=\; 0.95208 \; \Longrightarrow \;0.95... ... \;\\ [1ex] C &=& 0.95208 \,\mbox{e}^{-1.0932^2} \;=\; 0.28817 \end{eqnarray*}$$
   
   
   
   
   
   which we enter into init.m.

Completing Items 7-10, we obtain the finished product illustrated on the next page. (Note: your work in the upper left will be neatly hand written, not typed.)