Math307
Modeling with ODEs
A Few Pointers
Often times, when working with story problems, or with excape velocities or the like, you will see what looks like too many variables, and may be tempted to take a definite integral from some constant to {$x$} and solve for {$x$}. Not only is this asking for sign errors, but there’s a much easier way. Recall that
{$ \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = \frac{dv}{dx}v $}
Using this substitution, one can eliminate {$t$} from an equation that would otherwise be pretty damn hairy.