9.3 Solution Methods for ordinary Differential Equations

If one wants to obtain from the differential equation a closed form for y(t), then it needs to be solved analytically, as shown for the particularly easy case of the exponential function. If the function y does not appear on the right hand side of the differential equation, i.e. if the differential equation reads y = f(t), then y is obtained via the normal integration. We can immediately verify this for the traveled distance problem. For constant acceleration b we obtain for the traveled distance s(t) the following:

s = b;s = v =0tbdt = b0tdt = bt+v 0;s =0t(bt + v 0)dt = bt2 2 +v0t+s0

Here the two initial values are the initial position s0 and the initial velocity v0.

For the general case the art of solving differential equations analytically fills whole books. The solution methods for those differential equations, that are import in Physics mostly follow simple patterns, for which there standard methods. We refer here to a few of the books cited in the introduction. In general all ordinary differential equations can be treated analytically. For this endeavor an approach quite similar to the integration of non-standard functions is applied: One tries to guess a special solution systematically and then tries to obtain a general solution with the variation or determination of parameters, which can be an exact, an approximate or a series solution of the differential equation.

If one does not have to obtain the solution as an analytical expression, but can be satisfied with calculating its numerical values as a function of the initial values and variables and thus also to represent its general behavior graphically, then one can solve the differential equation numerically, irrespective of its complexity. All popular programs like Mathematica or Java/EJS provide a number of methods with different degrees of accuracy, that can be easily used. However the algorithms use stay hidden in the background ( black box), which is why we will be describing and visualizing the most important ones in the following.

In practice it is quite important, to become familiar with the numerical methods of solution for differential equations of first order, since all other ordinary differential equations can be reduced to them, if one allows several dependent variables. We visualize this in the following for equations of first order and also show in detail the application to differential equations of second order. All following extensions work in a similar manner.