The useful application domain for L-transforms is normally considered to be restricted to LTI…

The useful application domain for L-transforms is normally
considered to be restricted to LTI systems. But the solution of some
time-varying differential equations is also simpler in the frequency domain. An
example is Bessel’s equation

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a) Use the Differentiation and Multiply-by-t Theorems to
transform this equation into a first-order differential equation in Xes) =
L[x(t)] .

b) The equation in Xes) can be integrated by separation of
variables. Show (by substitution) that a solution is

where C is an arbitrary constant.

c) The zero-order Bessel function x(t) = Jo(t) is a solution
of the original equation. Indeed, the irrational function obtained in (b)-with
an appropriate choice of Cis L [Jo(t») . Jo(O) is normally defined to be 1;
what is the appropriate choice of C?

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