Translations:Help:SVG for math/6/en

Solutions to Differential Equations

Trajectories of a cast projectile under different initial conditions.

The plot above gives the solutions of the following differential equations under different initial conditions.

$y''(t)=-g,\quad x''(t)=0,$

$x'(t)=v_{0}\cos \theta ,\quad y'(t)=v_{0}\sin \theta ,\quad x(0)=y(0)=0.$

The solutions to this differential system have simple closed forms. Hence the plotting program only needs routine computations. As a result, one may use Gnuplot for this case. To plot the multiple curves in the graph, we can leverage the for loop in Gnuplot (version 4.4+ required). This technique can also produce the plots of some elementary functions (exponential, logarithm, etc.).