This secton provides a stream of examples demonstating applications of power
series method for solving initial value problems for second order differential
equations.
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where p(x) and q(x) are holomorphic functions at the origin, which is a regular singular point for Eq.\eqref{EqComplex.1}. Here \( y' = {\text d}y/{\text d}x , \quad y'' = {\text d}^2 y/{\text d}x^2 \) are derivatives with respect to x, the independent avriable. The standard method of solution---Fuch--Frobenius method---consists in assuming a series solution of the form
where the exponent r is chosen so that the leading coefficient c_{0}
is non-zero,
substituting into equation (1), and determining the coefficients c_{k} recursively.
The condition c_{0} ≠ 0 leads to the so-called indicial equation
(a quadratic equation with real coefficients) whose roots determine r.
In this section, we discuss the general equation \eqref{EqComplex.1} where the roots of the indicial equation are complex.
Theorem 1:
If the indicial equation corresponding to the series solution \eqref{EqComplex.2} of the differential equation \eqref{EqComplex.1} has complex roots, then the real and imaginary parts of Eq.\eqref{EqComplex.2} are two real independent solutions of Eq.\eqref{EqComplex.1} and hence constitute a basis for the solution space.
Suppose that y(x) = u(x) + jv(x) is a complex non-trivial solution of
equation (1), where u(x) and v(x) are real non-zero functions. Substituting
into equation (1), remembering that the coefficients of the equation are real
functions, and setting real and imaginary parts equal to zero, we obtain
Hence, u(x) and v(x) are
two real functions satisfying the differential equation. It remains to prove that they are linearly independent. Suppose, for example, that the root
r = α + jβ (the other root is r = α − jβ) is used in generating the complex solution y(x). Since c_{0} ≠ 0 (the required condition for generating r), the leading term in the complex series solution for y(x) is
Clearly u(x) and v(x) are linearly independent.
﹡ ⁎ ✱ ✲ ✳ ✺ ✻ ✼
✽ ❋
This theorem suggests that the best and perhaps fastest way of generating
the two real solutions is to use the Frobenius method to obtain just one complex
solution using any one of the two complex conjugate roots, and then taking real
and imaginary parts. It is also of interest to observe that, in a certain sense, the
treatment of the complex case is unlike that of the real root case. In the real root
case, the differential equation must be solved twice using the Frobenius method in
order to obtain the two independent solutions. In the complex case, the
differential equation is solved only once.
Let us consider a particular case when the coefficients of the differential equation \eqref{EqComplex.1} are two-term polynomials:
All the coefficients c_{k}, k = 1,2, …, m−1, equal zero since the coefficient of c_{k} reduces, on using the indicial equation, to
\( k^2 - 2k + p_0 k + 2kr , \) which is never zero since r has a non-zero imaginary part. Thus, with n = mp, the recurrence relation in the new index p reduces to
Letting c_{m0} = 1, and remembering that
\( \displaystyle x^{{\bf j}\beta} = \exp \left\{ {\bf j}\beta \ln x \right\} , \) the solution in complex
form becomes
Example 1:
Consider the differential equation
\[
x^2 y'' - x\,y' + \left( x + 2 \right) y = 0 .
\]
In this case, m = 1, p_{0} = −1, p_{1} = 0, q_{0} = 2, and q_{1} = 2. The roots of the indicial equation are
Grigorieva, E., Methods of Solving Sequence and Series Problems, Birkhäuser; 1st ed. 2016.
Neuringer, J., The Frobenius method for complex roots of the
indicial equation, International Journal of Mathematical Education in Science and Technology, 1978, Vol. 9, No. 1, pp. 71--77. https://doi.org/10.1080/0020739780090110
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