Difference Equations to Differential Equations (54).pdf

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 £© Section

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 £© Power Series Solutions
In this section we consider one more approach to finding solutions, or approximate so-
lutions, to differential equations. Although the method may be applied to first order
equations, our discussion will center on second order equations.
The idea is simple: Assuming that the equation
x¨ = f(x, x,˙ t) ()
has a solution which is analytic on an interval about t = t0, we express x as a power series
∞
x(t) = a (t − t )n, ()
X n 0
n=0
computex ˙ andx ¨, substitute the results into the equation, solve for the coefficients a0, a1,
a2, . . . , and verify that the resulting series converges on an interval about t0. As we shall
see, in practice the difficult part is solving for the coefficients. This method will lead us to a
closed form solution for the equation only in the rare case that we are able to recognize the
resulting power series as the Taylor series of some known function. One advantage of this
technique over numerical methods, such as the Runge-Kutta method, is that we are able
to work with general solutions and equations involving unspecified parameters, whereas
with a numerical method every quantity must be specified as a number. The disadvantage
of this technique is that it is not as widely applicable, due to the difficulty of solving for
the coefficients, and, when numerical results are needed, one must still approximate the
infinite series which results when evaluating x at a point.
To illustrate the procedure, we will begin with an example which we know to be solvable
by the techniques of Section .
Example Consider the equation
x¨ = −x. ()
This is a constant coefficient homogeneous linear equation with characteristic equation
k2 + 1 = 0. Since the roots of the characteristic equation are −i and i, we know from our
work in Section that the general solution of this equation is
x = c1 cos