Power Series Solutions of Differential Equations

The command to solve differential equations in power series around a particular initial point with specific initial conditions is called seriesSolve. It can take a variety of parameters, so we illustrate its use with some examples. Since the coefficients of some solutions are quite large, we reset the default to compute only seven terms.

You can solve a single nonlinear equation of any order. For example, we solve

      y''' = sin(y'') * exp(y) + cos(x)

subject to y(0)=1, y'(0)=0, y''(0)=0 We first tell Axiom that the symbol 'y denotes a new operator.

Enter the differential equation using y like any system function.

Solve it around x=0 with the initial conditions y(0)=1, y'(0)=y''(0)=0.

You can also solve a system of nonlinear first order equations. For example, we solve a system that has tan(t) and sec(t) as solutions. We tell Axiom that x is also an operator.

Enter the two equations forming our system.

Solve the system around t=0 with the initial conditions x(0)=0 and y(0)=1. Notice that since we give the unknowns in the order [x,y], the answer is a list of two series in the order [series for x(t), series for y(t)].

The order in which we give the equations and the initial conditions has no effect on the order of the solution.