Slope Fields for First and Second Order ODEs | Definition and Generation

[Jhenrique]

I know that the standard definition for a slope field is ##\frac{dy}{dx} = f(x, y)##, but and if the equation given is a second-order ODE ##a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=0## or a system of first-order ODEs ##A\frac{d\vec{r}}{dt}+\vec{b}=\vec{0}##, the definition for slope field continues the same? I need only isolate dy/dx and thus the slope field is automatically generated?

 

[HallsofIvy]

Given a second order differential equation, you would write it as two first order differential equations:
[itex]z= dy/dx[/itex] and [itex]adz/dx+ bz+ cy= 0[/itex].

Now, if your [itex]\vec{r}[/itex] has greater dimension than 1 you "slope field" would have to have greater dimensions also. With the single variable y as a function of t, your "slope field" is a two dimension graph with axes y and t, so can be drawn on a sheet of paper. If you have variables x, y as part of your vector function, your "slope field" is a three dimension graph with axes x, y, and t. If you have variables x, y, as part of your vector function, your slope field is a four dimension graph with axes x, y, z, and t. If you succeed in drawing such a thing, please post it here!