Unusual Bessel-like Differential Equation with Negative Sign

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SUMMARY

The forum discussion centers on solving an unusual Bessel-like differential equation that includes a negative sign on the dependent variable. The user attempts to transform the equation into a form resembling the Spherical Bessel differential equation by multiplying both sides by x². However, the presence of the negative sign complicates the transformation, leading to uncertainty about the next steps in solving the equation. A reference to the Modified Spherical Bessel Differential Equation is provided as a potential resource for further understanding.

PREREQUISITES
  • Understanding of differential equations, specifically Bessel functions.
  • Familiarity with the Spherical Bessel differential equation.
  • Basic knowledge of mathematical transformations and manipulations.
  • Access to mathematical resources such as Wolfram MathWorld.
NEXT STEPS
  • Research the properties and solutions of the Spherical Bessel differential equation.
  • Study the Modified Spherical Bessel Differential Equation for insights on handling negative signs.
  • Explore mathematical techniques for transforming differential equations.
  • Utilize computational tools like Mathematica to analyze and solve differential equations.
USEFUL FOR

Mathematicians, physics students, and anyone dealing with differential equations, particularly those interested in Bessel functions and their applications.

maistral
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Hi. I have this Bessel look-a-like DE (however it has the negative sign on the dependent variable):

2pql7i8.png


My attempt at a solution is by (obviously) trying to transform it into a Bessel DE by multiplying it with x2 on both sides, then it would look like the Spherical Bessel differential equation, except that there's that negative sign beside y.

Then what? :nb) I have no idea what to do. Can someone tell me what this DE is supposed to look like so I can solve it on my own?
 
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maistral said:
Hi. I have this Bessel look-a-like DE (however it has the negative sign on the dependent variable):

View attachment 203702

My attempt at a solution is by (obviously) trying to transform it into a Bessel DE by multiplying it with x2 on both sides, then it would look like the Spherical Bessel differential equation, except that there's that negative sign beside y.

Then what? :nb) I have no idea what to do. Can someone tell me what this DE is supposed to look like so I can solve it on my own?
http://mathworld.wolfram.com/ModifiedSphericalBesselDifferentialEquation.html
 
Got it. Thanks!
 

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