$\stackrel{\.}{x}\=y\,\stackrel{\.}{y}\=x\,x\left(0\right)\=a\,y\left(0\right)\=b$

The best interactive tool for integrating a gradient field graphically is the task template shown in Figure 9.3.4(b).  Bounds for the graph are established by entering values under the Plot Window heading. The components of the vector field are entered under the Vector Field heading. The coordinate system and coordinate-variable names are next given under the Coordinates heading. The name and range for the parameter along the integral curves are specified under the Path Parameter heading. Pressing the Enter Data button initializes the graph, which then displays arrows of the vector field. Click on the vector field, and the coordinates of the point clicked on will appear in the window at the bottom of the task template. The integral curve through that point will appear on the graph. In Figure 9.3.4(b), eight integral curves were so obtained, and the result resembles the graph in Figure 9.3.4(a).

Table 9.3.4(a) contains an interactive solution of the system of differential equations whose solutions are curves along which the gradient field is tangent.

Squaring both sides of either solution results in $g\left(x\,y\right)\equiv y^2-x^2\=b^2-a^2$.

An interactive solution of $y\prime \=x\/y$, the alternate differential equation, is obtained in Table 9.3.4(b).

Of course, squaring both sides of either solution leads to $y^2-x^2\=c$, where, for the solution to pass through $\left(a\,b\right)$, it must be that $c\=b^2-a^2$.

Alternatively, if the Context Panel is launched on $y\prime \=x\/y\,y\left(a\right)\=b$, and the option Solve DE Interactively is selected, then the  Assistant is launched, as shown in Figure 9.3.4(c). Pressing the Solve Symbolically button opens the second pane, whose upper half is shown in Figure 9.3.4(d). On the left, in the section labeled "Method", set the "Explicit" field to "no" so that Maple does not automatically try to provide an explicit solution. Hence, the implicit solution shown in the figure is returned when the Solve button under "Output" is pressed.$$

With slight re-arrangement, the solution $y^2-x^2\=b^2-a^2$ has again been obtained.

Invoking the dsolve command as per Table 9.3.4(c) returns the solution of the initial-value problem $y\prime \=x\/y\,y\left(a\right)\=b$. Note the inclusion of the implicit option.

Table 9.3.4(d) contains the much more tedious solution of the coupled equations $\stackrel{\.}{x}\=y\,\stackrel{\.}{y}\=x$. The equations are solved with the dsolve command, but $x\left(t\right)$ and $y\left(t\right)$ must be extracted separately with the eval command. The eliminate command is used to eliminate the parameter $t$, and the op command is used to select one of the two possible solutions. Finally, the solve command is applied to isolate $y$, from which it is again possible to obtain $y^2-x^2\=\mathrm{constant}$.