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This is a helpful document I got from one of my DE's teachers in graduate school, and I've toted it around with me. I will type it up here, as well as attach a pdf you can download.

$$J_{\nu}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{\nu+2m}}{2^{\nu+2m} \, m! \,\Gamma(\nu+m+1)}$$

is a

The solutions of $x^2 \, y''+x \, y'+(-x^2-\nu^2) \, y=0$ are expressible in terms of

The graphs:

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View attachment 4755

View attachment 4754

View attachment 4753

You can use these graphs sometimes to work out initial conditions, particularly if any of them are zero.

If $(1-a)^2\ge 4c$ and if neither $d$, $p$ nor $q$ is zero, then, except in the obvious special case when it reduces to the Cauchy-Euler equation $(x^2 y''+axy'+cy=0)$, the differential equation

$$x^2y''+x(a+2bx^p)y'+[c+dx^{2q}+b(a+p-1)x^p+b^2x^{2p}]y=0$$

has as general solution

$$y=x^{\alpha} \, e^{-\beta x^p} [C_1 \, J_{\nu}(\varepsilon x^q)+C_2 Y_{\nu}(\varepsilon x^q)]$$

where

$$\alpha=\frac{1-a}{2}, \qquad \beta=\frac{b}{p},\qquad \varepsilon=\frac{\sqrt{|d|}}{q},\qquad \nu=

\frac{\sqrt{(1-a)^2-4c}}{2q}.$$

If $d<0$, then $J_{\nu}$ and $Y_{\nu}$ are to be replaced by $I_{\nu}$ and $K_{\nu}$, respectively. If $\nu$ is not an integer, then $Y_{\nu}$ and $K_{\nu}$ can be replaced by $J_{-\nu}$ and $I_{-\nu}$ if desired.

The following file is a pdf of the above.

View attachment 4760

Bessel Functions

$$J_{\nu}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{\nu+2m}}{2^{\nu+2m} \, m! \,\Gamma(\nu+m+1)}$$

is a

*Bessel function of the first kind of order*$\nu$. The general solution of $x^2 \, y''+x \, y'+(x^2-\nu^2) \, y=0$ is $y=c_1 \, J_{\nu}(x)+c_2 \, J_{-\nu}(x)$. If $\nu=n$ is an integer, the general solution is $y=c_1 \, J_n(x)+c_2 \, Y_n(x)$ where $Y_n(x)$ is the*Bessel function of the second kind of order*$n$. Here, $Y_n(x)$ equals $\frac{2}{\pi} \, \ln\left(\frac{x}{s}\right)$ plus a power series.The solutions of $x^2 \, y''+x \, y'+(-x^2-\nu^2) \, y=0$ are expressible in terms of

*modified Bessel functions of the first/second kind of order*$\nu$, namely $I_{\nu}(x)$ and $K_{\nu}(x)$.The graphs:

View attachment 4758

View attachment 4757

View attachment 4756

View attachment 4755

View attachment 4754

View attachment 4753

You can use these graphs sometimes to work out initial conditions, particularly if any of them are zero.

Equations Solvable in Terms of Bessel Functions

If $(1-a)^2\ge 4c$ and if neither $d$, $p$ nor $q$ is zero, then, except in the obvious special case when it reduces to the Cauchy-Euler equation $(x^2 y''+axy'+cy=0)$, the differential equation

$$x^2y''+x(a+2bx^p)y'+[c+dx^{2q}+b(a+p-1)x^p+b^2x^{2p}]y=0$$

has as general solution

$$y=x^{\alpha} \, e^{-\beta x^p} [C_1 \, J_{\nu}(\varepsilon x^q)+C_2 Y_{\nu}(\varepsilon x^q)]$$

where

$$\alpha=\frac{1-a}{2}, \qquad \beta=\frac{b}{p},\qquad \varepsilon=\frac{\sqrt{|d|}}{q},\qquad \nu=

\frac{\sqrt{(1-a)^2-4c}}{2q}.$$

If $d<0$, then $J_{\nu}$ and $Y_{\nu}$ are to be replaced by $I_{\nu}$ and $K_{\nu}$, respectively. If $\nu$ is not an integer, then $Y_{\nu}$ and $K_{\nu}$ can be replaced by $J_{-\nu}$ and $I_{-\nu}$ if desired.

The following file is a pdf of the above.

View attachment 4760