Solving 2nd Order ODEs w/ Variable Coefficients

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SUMMARY

There is no general solution for second order homogeneous differential equations with variable coefficients. While specific forms, such as Euler's equation, can be explicitly solved, many others, including Bessel's equation, require specialized techniques for analysis. These techniques include studying asymptotic behavior, integral and differential identities, and infinite sum expansions. The ability to derive a second solution from a known first solution is a valuable strategy in tackling these equations.

PREREQUISITES
  • Understanding of second order linear ordinary differential equations (ODEs)
  • Familiarity with Bessel functions and their properties
  • Knowledge of asymptotic analysis techniques
  • Experience with differential identities and integral calculus
NEXT STEPS
  • Research the properties and applications of Bessel functions
  • Study asymptotic behavior in differential equations
  • Learn techniques for solving Euler's equations
  • Explore integral and differential identities related to ODEs
USEFUL FOR

Mathematicians, physicists, and engineering students who are dealing with second order differential equations and seeking to understand variable coefficients and their implications in various applications.

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I want to know if there is a general solution to a second order homogeneous differential equation with variable coefficients?
 
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There are only few forms of such equations that can be explicitly solved (for example Euler's equation)

Other equations can not be solved directly, but they are important enough to get their own names and to be studied with different tools to give some characteristics of them (asymptotic behaviour, integral and differential identities, infinite sum expansion) but cannot be explicitly written (for example Bessel's Equation and its solutions the Bessel Functions)

You can always know the second solution of an ODE by just knowing the first solution, so from times to times guessing can come in handy.

In general I believe there are many problems to study that arise from second order linear ODE, and you can acquire many tools to do so along the way. But the bottom line is that there is no general method of solving them.
 
thanks..great help
Was working with a friend on getting a general solution..
Has proven to be a real problem but we giving it more tries.
Thanks once again
 

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