# Linear Dependency, Wronskian and Bessel's Equation

Three problems regarding the Wronskian and solutions of a second order differential equation.

Example of a question

1. Determine whether the following sets of functions are linearly dependant or independent... Please see attached.

2. Bessel's equation xÂ²y" + xy' + (xÂ² - nÂ²)y = 0

where n is a constant, is an important differential equation that arises, for example, in the study of the vibrations of a circular drumhead.

a) Let y1(x) and y2(x) be any two solutions to Bessel's equation. Without computing y1 and y2, determine their Wronskian.

b) One solution to (1) with n=1/2 is sinx/sqrt(x) Find a second linearly independent solution.

c) Compute the Wronskian for the two solutions you found in part (b).

Please see attached for full question.

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#### Solution Preview

Hi there!

The solution is attcahed below in two formats (the pdf is for the ...

#### Solution Summary

The solutions (8 pages long) show how to incorporate the Wronskian and the solutions of second order ODE's.

The solution also shows hwo to arrive at Abel's identity and its implications.