Question

Discuss how you would define a singular point for the linear third-order differential equation.

${\mathbf{a}}_{\mathbf{3}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}{\mathbf{a}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{y}\mathbf{\text{'}}\mathbf{+}{\mathbf{a}}_{\mathbf{0}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{y}\mathbf{=}\mathbf{0}$

Short answer

The third order differential equation has a regular singular point at x = x₀if the following functions are analytical at x = x₀.

Step-by-step solution

Step 1:

Our aim is to develop some sort of method, upon which we can define the singular points of the third order differential equation. First, we will consider the second order case and generalize the definition the definition to the third order one. Let’s assume we have a second order differential equation with the following form

${\mathrm{a}}_3\left(\mathrm{x}\right)\mathrm{y}\text{'}\text{'}\text{'}+{\mathrm{a}}_2\left(\mathrm{x}\right)\mathrm{y}\text{'}\text{'}+{\mathrm{a}}_1\left(\mathrm{x}\right)\mathrm{y}\text{'}+{\mathrm{a}}_0\left(\mathrm{x}\right)\mathrm{y}=0$

$\begin{array}{rcl}{\mathrm{a}}_1\left(\mathrm{x}\right)\mathrm{y}\text{'}\text{'}+{\mathrm{a}}_2\left(\mathrm{x}\right)\mathrm{y}\text{'}+{\mathrm{a}}_3\left(\mathrm{x}\right)\mathrm{y}& =& 0\\ \mathrm{y}\text{'}\text{'}+\frac{{\mathrm{a}}_2\left(\mathrm{x}\right)}{{\mathrm{a}}_1\left(\mathrm{x}\right)}\mathrm{y}\text{'}+\frac{{\mathrm{a}}_3\left(\mathrm{x}\right)}{{\mathrm{a}}_2\left(\mathrm{x}\right)}\mathrm{y}& =& 0\\ \mathrm{y}\text{'}\text{'}+\mathrm{P}\left(\mathrm{x}\right)\mathrm{y}\text{'}+\mathrm{Q}\left(\mathrm{x}\right)\mathrm{y}& =& 0...\left(1\right)\end{array}$

Where

$\mathrm{P}\left(\mathrm{x}\right)=\frac{{\mathrm{a}}_2\left(\mathrm{x}\right)}{{\mathrm{a}}_1\left(\mathrm{x}\right)},\mathrm{Q}\left(\mathrm{x}\right)=\frac{{\mathrm{a}}_3\left(\mathrm{x}\right)}{{\mathrm{a}}_1\left(\mathrm{x}\right)}...\left(2\right)$

Step 2:

Equation (1) is the standard form, upon which we will build our analysis. We recall the following definition.

Point x =x₀ is said to be an ordinary point of the differential equation, if P(x)Q(x) are analytical at that point, otherwise x₀is said to be a singular point.

Step 3:

Then, we can classify this singular point according to the following definition.

A singular pointx₀ is said to be a regular singular point of the differential equation, if P(x)Q(x) are analytical at x₀, otherwise it is irregular singular point.

Where P(x)Q(x) is defined as follows

$$\begin{gathered} \mathrm{p}\left(\mathrm{x}\right)=(\mathrm{x}-{\mathrm{x}}_0)\mathrm{P}\left(\mathrm{x}\right) \\ \mathrm{q}\left(\mathrm{x}\right)=(\mathrm{x}-{\mathrm{x}}_0{)}^2\mathrm{Q}\left(\mathrm{x}\right) \end{gathered}$$

Step 4:

Now, we put the given differential equation on the standard for.

$\begin{array}{rcl}\frac{{\mathrm{a}}_3\left(\mathrm{x}\right)}{{\mathrm{a}}_3\left(\mathrm{x}\right)}\mathrm{y}\text{'}\text{'}\text{'}+\frac{{\mathrm{a}}_2\left(\mathrm{x}\right)}{{\mathrm{a}}_3\left(\mathrm{x}\right)}\mathrm{y}\text{'}\text{'}+\frac{{\mathrm{a}}_1\left(\mathrm{x}\right)}{{\mathrm{a}}_3\left(\mathrm{x}\right)}\mathrm{y}\text{'}+\frac{{\mathrm{a}}_0\left(\mathrm{x}\right)}{{\mathrm{a}}_3\left(\mathrm{x}\right)}\mathrm{y}& =& 0\\ \mathrm{y}\text{'}\text{'}\text{'}+\mathrm{P}\left(\mathrm{x}\right)\mathrm{y}\text{'}\text{'}+\mathrm{Q}\left(\mathrm{x}\right)\mathrm{y}\text{'}+\mathrm{G}\left(\mathrm{x}\right)\mathrm{y}& =& 0\end{array}$

$\mathrm{P}\left(\mathrm{x}\right)=\frac{{\mathrm{a}}_2\left(\mathrm{x}\right)}{{\mathrm{a}}_3\left(\mathrm{x}\right)},\mathrm{Q}\left(\mathrm{x}\right)=\frac{{\mathrm{a}}_1\left(\mathrm{x}\right)}{{\mathrm{a}}_3\left(\mathrm{x}\right)},\mathrm{G}\left(\mathrm{x}\right)=\frac{{\mathrm{a}}_0\left(\mathrm{x}\right)}{{\mathrm{a}}_3\left(\mathrm{x}\right)}$

Then, according to the analysis above, we say that the third order differential equation has a regular singular point atx =x₀ if the following functions are analytic at x =x₀.

$$\begin{gathered} \mathrm{p}\left(\mathrm{x}\right)=(\mathrm{x}-{\mathrm{x}}_0)\mathrm{P}\left(\mathrm{x}\right) \\ \mathrm{q}\left(\mathrm{x}\right)=(\mathrm{x}-{\mathrm{x}}_0{)}^2\mathrm{Q}\left(\mathrm{x}\right) \\ \mathrm{g}\left(\mathrm{x}\right)=(\mathrm{x}-{\mathrm{x}}_0{)}^3\mathrm{G}\left(\mathrm{x}\right) \end{gathered}$$

The third order differential equation has a regular singular point atx =x₀ if the following functions are analytical atx =x₀.