📂Odinary Differential Equations

Frobenius Method

Explanation¹

There are various methods to solve differential equations. One of them involves assuming the solution as a power series, as follows.

$$ y=\sum \limits_{n=0}^{\infty} a_{n}x^{n} $$

However, some series cannot be represented in the form above. For example, as follows.

$$ \frac{\cos x}{x^{2}}=\frac{1}{x^{2}}-\frac{1}{2!}+\frac{ x^{2}}{4!}-\cdots $$

$$ \sqrt{x} \sin x = x^{\frac{1}{2}}\left( x - \frac{x^{3}}{3!}+\cdots \right) $$

In such cases, the solution is assumed to be in the following form.

$$ \begin{equation} y=\sum \limits_{n=0}^{\infty} a_{n} x^{n+r}=x^{r}\sum \limits_{n=0}^{\infty} a_{n}x^{n} \label{eq1} \end{equation} $$

In this case, $r$ can be positive, negative, or even a rational number. Also, if $a_{0}=0$, the first term changes, so it is always assumed to be $a_{0}\ne 0$. The series $\eqref{eq1}$ is called a generalized power series. The method of solving differential equations by assuming the solution as a generalized power series is called the Frobenius method. An example of using the Frobenius method is the Bessel equation.