|8.2 Section||2 Background .1 Cosmology .11 Math||Section 8.4|
Section 3 Solution by undetermined coefficients
This method can also be used to solve linear differential equations with variable coefficients. Its application is explained by the example of the second order equation
with the initial conditions y(0) = y0, y'(0) = y'0? Assuming that the coefficient function in (3.1) can be expanded in power series
search for the solution of the given equation in the form
where the cn must be determined. Differentiating (3.2) twice with respect to x:
and substituting these series into (3.1), one finds
Expanding the products of the series and equating the coefficients of equal powers of x on the left- and right-hand sides, one arrives at the system of equations
where denotes a linear function of
Consecutive equations of the system (3.4) contain
one additional unknown. The coefficients c0 and c1 are determined by the initial
conditions, while all remaining
ones are obtained from (2.12). It can be shown that, if the series
converge for |x| < R, then the series obtained converges in the same interval.
Note: If the initial conditions are given for x = x0, then it is advisable to make the substitution x - x0 = t, thus reducing the problem to the one considered above.
Example 1: Find the solution of the equation
for the initial conditions y0) = 0, y'(0) = 1.
Solution: Expand the coefficients of the given equation in power series.
Find the solution of (3.5) in the form of the power series
Substituting these series into (3.5) and equating the coefficients of equal powers of x, one finds the system for the determination of the coefficients ci:
By the initial conditions, c0 = 0, c1 = 1.It is readily seen that c2n+1 = 0, n = 1, 2, . . . , so that
The solution of the problem has thus the form
Sometimes, when solving a differential equation by the method
of undetermined coefficients, one may find an expression for the
coefficients of a
series in a general form, as is illustrated by
Example 5: Find the power series solution of the equation
for the initial conditions y(0)=5, y'(0)=2.
Solution: Look for the solution in the form of the series
Substituting these series into(3.6) and equating coefficients of equal powers of x on both sides leads to the system:
The initial conditions yield c0 = 5 and c1 = 2. The solution of (3.7) is
Separating coefficients with odd and even subscripts:
Substituting for cl and c2, one finds
Separate the series into even and odd powers of x:
Next, find the domain of convergence of these series. For the first series, the limit of the absolute value of the ratio of consecutive terms is
for the second series:
It is seen that both series converge for all values of x, whence the solution of (3.7) can be written in the form:
Using undetermined coefficients, find the solutions of the equations :