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 Jan 26, 2012
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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 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 Y1Plot.pdf
View attachment K1Plot.pdf
View attachment I1Plot.pdf
View attachment I0Plot.pdf
View attachment J1Plot.pdf
View attachment J0Plot.pdf
You can use these graphs sometimes to work out initial conditions, particularly if any of them are zero.
If $(1a)^2\ge 4c$ and if neither $d$, $p$ nor $q$ is zero, then, except in the obvious special case when it reduces to the CauchyEuler equation $(x^2 y''+axy'+cy=0)$, the differential equation
$$x^2y''+x(a+2bx^p)y'+[c+dx^{2q}+b(a+p1)x^p+b^2x^{2p}]y=0$$
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
$$\alpha=\frac{1a}{2}, \qquad \beta=\frac{b}{p},\qquad \varepsilon=\frac{\sqrt{d}}{q},\qquad \nu=
\frac{\sqrt{(1a)^24c}}{2q}.$$
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 Bessel Function Cheat Sheet.pdf
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 Y1Plot.pdf
View attachment K1Plot.pdf
View attachment I1Plot.pdf
View attachment I0Plot.pdf
View attachment J1Plot.pdf
View attachment J0Plot.pdf
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 $(1a)^2\ge 4c$ and if neither $d$, $p$ nor $q$ is zero, then, except in the obvious special case when it reduces to the CauchyEuler equation $(x^2 y''+axy'+cy=0)$, the differential equation
$$x^2y''+x(a+2bx^p)y'+[c+dx^{2q}+b(a+p1)x^p+b^2x^{2p}]y=0$$
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
$$\alpha=\frac{1a}{2}, \qquad \beta=\frac{b}{p},\qquad \varepsilon=\frac{\sqrt{d}}{q},\qquad \nu=
\frac{\sqrt{(1a)^24c}}{2q}.$$
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 Bessel Function Cheat Sheet.pdf
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