A homogeneous second order linear differential equation takes the form of
$$a_0(x)\frac{d^2y}{dx^2}+a_1(x)\frac{dy}{dx}+a_2(x)y=0 , a_0(x)\neq 0$$
Dividing through $a_0(x)$
$$\frac{d^2y}{dx^2}+P_1(x)\frac{dy}{dx}+P_2(x)y=0$$
Where $$P_1(x)=\frac{a_1(x)}{a_0(x)},P_2(x)=\frac{a_2(x)}{a_0(x)}$$
The Frobenius method is defined as such.
- Suppose that $x_0$ is a regular singular point of the second order homogeneous differential equation. And, we have a solution that takes the form of
$$y=(x-x_0)^r\sum_{n=0}^\infty c_n(x-x_0)^n,c_0 \neq 0$$
From other perspective we have
$$y=\sum_{n=0}^\infty c_n(x-x_0)^{n+r},c_0 \neq 0$$
- Assume that the differentiation term by term is valid.
$$\frac{dy}{dx}=\sum_{n=0}^\infty(n+r) c_n(x-x_0)^{n+r-1}$$
$$ \frac{d^2y}{dx^2}=\sum_{n=0}^\infty(n+r)(n+r-1) c_n(x-x_0)^{n+r-2}$$
- Substituting all the series into the Differential equation. Hence, we have
$$K_0(x-x_0)^{n+r}+K_1(x-x_0)^{n+r-1}+K_2(x-x_0)^{n+r-2}+...=0$$
where $K_j(j=0,1,2,3..)$ are certain functions of r and the coefficients too.
- To make them valid we must set everything to zeroes.
$$K_0=K_1=K_2=...=0$$
Equating $K_0$ of the lowest power of $r+x$ to zeroes to obtain the unknown r. It is through solving a quadratic equation also known as incidental equation and the value of r we obtain is called the exponents of the function. We would have values of r such that $Re(r_1)\geq Re(r_2)$
Then, we equate $K_1=K_2=...=0$ we are lead to a set of condition that must be satisfied by the coefficients $c_n$ of the series.
7.Plug in $r_1$ into the above mentioned condition. We choose $c_n$ that satisfy the condition.
- If $r_2 \neq r_1$ we must repeat Step 7.
Questions.
1.Why in the first place our $c_0 \neq 0$ and why do we assume that? Does that imply that other values like $c_1,c_2,...$ can take zeroes What will happen if our $c_0$ is zeroes?
Secondly, why it is quite different from series method? I mean in series method when we take the derivative of the series the series then start at $n=1$. And taking the second derivative again the values start from $n=2$. In Fronenius method, we did not even change the index of the summation?
Why it is called deleted interval $0<|x-x_0|<R$? Under what condition it disappears from a normal second order linear homogeneous differential equation.
Can Frobenius method be applied to higher order like 3? If it does, we will have the three values of exponents r?