Differential equations – Series solutions of linear differential equations – The method of Frobenius

Theorem: If \(x_0\) is a regular singular point of the differential equation

\[y''(x)+p(x)y'(x)+q(x)y(x)=0,\]

then both \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\) are analytic at \(x=x_0\). That means that there exist positive constants \(R_1\) and \(R_2\) such that

\[(x-x_0)p(x)=\sum_{n=0}^{\infty}p_n(x-x_0)^n,\quad|x-x_0| < R_1\;\quad\text{and}\quad\;(x-x_0)^2q(x)=\sum_{n=0}^{\infty}q_n(x-x_0)^n,\quad|x-x_0| < R_2.\]

Then there exist solutions \(y(x)\) of the differential equation which can be written as

\[y(x)=|x-x_0|^r\sum_{n=0}^{\infty}c_n(x-x_0)^n,\quad c_0\neq0.\]

In fact, if \(R=\min\{R_1,R_2\}\) then we have

\[y(x)=|x-x_0|^r\sum_{n=0}^{\infty}c_n(x-x_0)^n,\quad 0 < |x-x_0| < R.\]

Here \(r\) is a solution of the indicial equation

\[r(r-1)+p_0r+q_0=0.\]

Moreover, if \(r_1,r_2\in\mathbb{C}\) are the solutions of this indicial equation with \(\text{Re}(r_1)\geq\text{Re}(r_2)\), then we have:

1. If \(r_1-r_2\notin\{0,1,2,\ldots\}\), then we have: \[y_1(x)=|x-x_0|^{r_1}\sum_{n=0}^{\infty}a_n(x-x_0)^n,\quad a_0\neq0\;\quad\text{and}\quad\;y_2(x)=|x-x_0|^{r_2}\sum_{n=0}^{\infty}b_n(x-x_0)^n,\quad b_0\neq0\]

   are two linearly independent solutions of the differential equation.



2. If \(r_1-r_2=0\), so \(r_1=r_2=r\), then we have: \[y_1(x)=|x-x_0|^{r}\sum_{n=0}^{\infty}a_n(x-x_0)^n,\quad a_0\neq0\;\quad\text{and}\quad\;y_2(x)=y_1(x)\ln|x-x_0|+|x-x_0|^{r}\sum_{n=0}^{\infty}b_n(x-x_0)^n\]

   are two linearly independent solutions of the differential equation.



3. If \(r_1-r_2\notin\{1,2,3,\ldots\}\), then we have: \[y_1(x)=|x-x_0|^{r_1}\sum_{n=0}^{\infty}a_n(x-x_0)^n,\quad a_0\neq0\;\quad\text{and}\quad\;y_2(x)=Ay_1(x)\ln|x-x_0|+|x-x_0|^{r_2}\sum_{n=0}^{\infty}b_n(x-x_0)^n,\quad b_0\neq0\]

   are two linearly independent solutions of the differential equation, where \(A\) is a constant that might be zero or not.

Some remarks:

• A series of th form \(\displaystyle|x-x_0|^r\sum_{n=0}^{\infty}c_n(x-x_0)^n\) is called a generalized power series. Only if \(r\in\{0,1,2\ldots\}\) this generalized power series reduces to an ordinary power series.


• The solutions \(r_1\) and \(r_2\) of the indicial equation are called indices.


• In the second case \(y_2(x)\) always has a logarithmic singularity. It is always possible to choose \(b_0=0\).


• In the third case \(y_2(x)\) only has a logarithmic singularity if \(A\neq0\), not if \(A=0\).

The proof of this theorem of Frobenius is constructive:

Proof: Multiply the differential equation by \((x-x_0)^2\):

\[(x-x_0)^2y''(x)+(x-x_0)^2p(x)y'(x)+(x-x_0)^2q(x)y(x)=0.\]

Now we have: \((x-x_0)p(x)=\displaystyle\sum_{n=0}^{\infty}p_n(x-x_0)^n\) and \((x-x_0)^2q(x)=\displaystyle\sum_{n=0}^{\infty}q_n(x-x_0)^n\) en dus

\[(x-x_0)^2y''(x)+(x-x_0)\left(\sum_{n=0}^{\infty}p_n(x-x_0)^n\right)y'(x)+\left(\sum_{n=0}^{\infty}q_n(x-x_0)^n\right)y(x)=0.\]

Now we substitute \(y(x)=(x-x_0)^r\displaystyle\sum_{n=0}^{\infty}c_n(x-x_0)^n=\sum_{n=0}^{\infty}c_n(x-x_0)^{n+r}\) with \(c_0\neq0\). Then we have:

\[y'(x)=\sum_{n=0}^{\infty}(n+r)c_n(x-x_0)^{n+r-1}\quad\text{and}\quad y''(x)=\sum_{n=0}^{\infty}(n+r)(n+r-1)c_n(x-x_0)^{n+r-2}.\]

Hence we have:

\begin{align*} &\sum_{n=0}^{\infty}(n+r)(n+r-1)c_n(x-x_0)^{n+r}+\left(\sum_{n=0}^{\infty}p_n(x-x_0)^n\right)\left(\sum_{n=0}^{\infty}(n+r)c_n(x-x_0)^{n+r}\right)\\[2.5mm] &{}\hspace{25mm}{}+\left(\sum_{n=0}^{\infty}q_n(x-x_0)^n\right)\left(\sum_{n=0}^{\infty}c_n(x-x_0)^{n+r}\right)=0. \end{align*}

This is only true if the coefficients of all powers of \(x-x_0\) are equal to zero. The coefficient of the lowest power of \(x-x_0\), that is \((x-x_0)^r\), is equal to

\[\{r(r-1)+p_0r+q_0\}c_0.\]

Since \(c_0\neq0\) this implies that \(r(r-1)+p_0r+q_0=0\) and that is the indicial equation. Further details are left out here.

Compare with the theory of Euler differential equations.

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Last modified on May 13, 2021