When you specify an initial condition [math](x_0,y_0)[/math] for the solution of a differential equation [math]y'=f(x,y)[/math], the [i]solution curve[/i] will pass through [math](x_0,y_0)[/math] with a slope of [math]f'(x_0,y_0)[/math]. These dynamic figures illustrate these slopes graphically with short line segments at selected points in the [math]xy[/math]-plane. Move the point in each figure to explore different curves.
[math]y'=y-x[/math][math][/math]
[math]y'=-\frac{2xy}{1+x^2}[/math]
[math]y'=\left(1-x\right)^2-y[/math]