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Introduction to Linear Algebra with Mathematica

Glossary

Non-standard Bessel equation

Some differential equations encountered in engineering, despite of their similarity in appearance to Bessel's equations, are not of the standard forms. Such differential equations can be solved after they are transformed into standard Bessel's equations via proper variable substitutions. It is known that the general solutions of the standard Bessel's equation of order ν:

\begin{equation} \label{EqNon.1} x^2 \frac{{\text d}^2 y}{{\text d} x^2} + x\,\frac{{\text d}y}{{\text d}x} + \left( x^2 - n^2 \right) y(x) = 0 \end{equation}
can be expressed as a linear combination of Bessel functions of first order Jn(x) and second order Yn(x)
\[ y(x) = c_1 J_{\nu} (x) + c_2 Y_{\nu} (x) , \]
where c₁ and c₂ are arbitrary constants. This statement was generalized by Bowman who showed that the non-standard Bessel's differential equation
\begin{equation} \label{EqNon.2} x^2 y'' + \left( 1 - 2\alpha \right) x\,y' + \left( \beta^2 \gamma^2 x^{2\gamma} + \alpha^2 - \nu^2 \gamma^2\right) y = 0 \end{equation}
has the general solution
\begin{equation} \label{EqNon.3} y(x) = c_1 x^{\alpha} J_{\nu} \left( \beta\,x^{\gamma} \right) + c_2 x^{\alpha} Y_{\nu} \left( \beta\,x^{\gamma} \right) \end{equation}
A special case of \eqref{EqNon.2}, when α = 0, is given by
\begin{equation} \label{EqNon.4} x^2 y'' + x\,y' + \left( \left( \beta\gamma x^{\gamma} \right)^2 - \nu^2 \gamma^2 \right) y = 0 , \end{equation}
whose general solution is reduced to be
\begin{equation} \label{EqNon.5} y(x) = c_1 J_{\nu} \left( \beta\,x^{\gamma} \right) + c_2 Y_{\nu} \left( \beta\,x^{\gamma} \right) . \end{equation}
Here, we refer (4) as non-standard Bessel's equation of second type while (2) is non-standard Bessel's equation of first type.    
Example 1: Let us consider the differential equation \[ x^2 y'' + x\,y' + \left( \frac{x}{4} - \frac{1}{16} \right) y = 0 . \] Observe that \( \displaystyle \quad \frac{x}{4} - \frac{1}{16} = \left( \frac{\sqrt{x}}{2} \right)^2 - \left( \frac{1}{4} \right)^2 , \quad \) one has \[ x^2 y'' + x\,y' + \left( \left( \frac{\sqrt{x}}{2} \right)^2 - \left( \frac{1}{4} \right)^2 \right) y = 0 . \] This equation is transferring into (1) if \[ z = \sqrt{x} , \qquad \nu = \frac{1}{2} . \] Hence, the general solution becomes \[ y(x) = c_1 J_{1/2} \left( \sqrt{x} \right) + c_2 Y_{1/2} \left( \sqrt{x} \right) . \] We check with Mathematica:
DSolve[x^2 *y''[x] + x*y'[x] + (x/4 - 1/16)*y[x] == 0, y[x], x]
{{y[x] -> (E^(I Sqrt[x]) C[1])/x^(1/4) + (I E^(-I Sqrt[x]) C[2])/x^( 1/4)}}

Moreover, consider another non-standard Bessel's equation of first type \[ x^2 y'' - 3x\,y' + 4 \left( x^4 - 3 \right) y = 0 . \] We can rewrite it as \[ x^2 u'' + \left( 2\delta -3 \right) x\,y' + \left( 4 x^4 - \left( \delta^2 - 4\delta - 12 \right) \right) u = 0 . \] Setting δ = 2, we obtain \[ x^2 u'' + x\,u' + \left( \left( 2 x^2 \right)^2 - 4^2 \right) u = 0 , \] which yields \[ y(x) = c_1 x^2 J_2 \left( x^2 \right) + c_2 x^2 Y_2 \left( x^2 \right) . \] We check with Mathematica:

DSolve[x^2 *y''[x] - 3* x*y'[x] + 4*(x^4 - 3)*y[x] == 0, y[x], x]
{{y[x] -> x^2 BesselJ[2, x^2] C[1] + I x^2 BesselY[2, x^2] C[2]}}
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End of Example 1
   
Example 2: We consider the differential equation \[ x^2 y'' + x\, y' + 9 x^4 y = 0 . \] Since \( \displaystyle \quad 9 x^4 = \left( 3\,x^2 \right)^2 - 0^2 , \quad \) we have \[ x^2 y'' + x\, y' + \left[ \left( 3\,x^2 \right)^2 - 0^2 \right] y = 0 . \] Therefore, the general solution of the given differential equation becomes \[ y(x) = c_1 J_0 \left( \frac{3}{2}\, x^2 \right) + c_2 Y_0 \left( \frac{3}{2}\, x^2 \right) . \] We check with Mathematica:
DSolve[x^2 *y''[x] + x*y'[x] + 9*x^4 *y[x] == 0, y[x], x]
{{y[x] -> BesselJ[0, (3 x^2)/2] C[1] + 2 BesselY[0, (3 x^2)/2] C[2]}}
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End of Example 2
   
Example 3: Let us consider the famous Airy's equation \[ y'' - x\,y = 0 . \] We set y = uxδ, then we get \[ x^2 u'' + 2\delta x\,u' - \left[ x^3 -\delta \left( \delta -1 \right) \right] u = 0 . \] If we set δ = ½, we reduce it to \[ x^2 u'' + x\,u' + \left( x^3 + \frac{1}{4} \right) u = 0 . \] Next, we rewrite it as \[ x^2 u'' + x\,u' + \left[ \left( {\bf j}\,x^{1/2} \right)^2 - \left( \frac{1}{2} \right)^2 \right] u = 0 , \] where j is the imaginary unit on the complex plane ℂ, so j² = −1. The general solution of the Airy equation becomes \[ y(x) = c_1 \sqrt{x} \, J_{1/3} \left( \frac{2}{3}\,{\bf j}\,x^{3/2} \right) + c_2 \sqrt{x} \, Y_{1/3} \left( \frac{2}{3}\,{\bf j}\,x^{3/2} \right) \] We check with Mathematica:
DSolve[y''[x] - x*y[x] == 0, y[x], x]
{{y[x] -> AiryAi[x] C[1] + AiryBi[x] C[2]}}
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End of Example 3

 

 

  1. Bowman, F., Introduction to Bessel Functions. Dover Publication Inc., New York, 1958.
  2. Bayman, B. F., “A generalization of the spherical harmonic gradient formula”, J. Math. Phys. 19 (1978), 2558–2562
  8. Watson, G.N., A Treatise on the Theory of Bessel Functions, Cambridge University Press; 2nd edition (August 1, 1995). ISBN-13 ‏ : ‎ 978-0521483919

 

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