A note about the sinc function as a solution to a differential equation

The (infamous , or famous , take your pick) sinc function is known to be one of two solutions of the differential equation:

differential equation with sinc function as a solution

This is a linear second order ordinary differential equation with dependent variable x and independent variable y.
I have tried to explore and find the solutions to this differential equation using mostly computer math software and programs.

The Texas Instruments 92 Plus scientific calculator and the Maple computer algebra system agree and give the same solution :

sinc solution to differential equationMathematica gives the following solution:

solution with expAfter converting the exponentials to trigonometric functions the expression above and the solution of the differential equation given by Mathematica becomes:

 Mathematica sinc solution

This solution is less simple than (1). Note that if we make the assumption in Mathematica that λ > 0 ,we get the solution:

solution with lambda positiveIf we try to find the graphical representation of solution (1) for different values of the arbitrary constants c1 and c2 and with λ (or n ) between -3 and 3 ,we get the following graphs:

sinc 2d graphs solution

Then I tried to take the absolute value of λ in the numerator of (2) for different values of the arbitrary constants , and compared their graphs  with the graphs of solution (1) for the same values of the constants. λ in (2) and n in (1) are between 1 and 4.graphs of sinc solution of differential equation

The graphical results are similar but not exactly the same.

Using the Manipulate built-in function in Mathematica , here is an animation of the graphs for solution (1) obtained by varying λ , c1 and c2:

Finally , I will give the solution of a generalized form of the differential equation above

general form of differential equation

obtained with Mathematica :

solution of generalized differential equationIn the solution above , Jμ(x) is the Bessel function of the first kind , and Yμ(x) is the Bessel function of the second kind.


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