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In this article, the solutions of Bessel's differential equations (DEs) by variable change method are formulated. To do so, we have considered the first and second kind of Bessel's functions which are obtained as solutions of Bessel's equations and it is used to determine the solutions of the lengthening pendulum (LP). To solve the given equations, we have used Frobenius theorem and the gamma function and hence, apply the obtained results to solve the LP. The finding reveals that Bessel's functions establish the solutions of LP equations. The solutions obtained for lengthening the pendulum are illustrated graphically using the computer software of MathLab. The graphical results show that the sinusoidal wave natures are compressed or extended based on the chosen parameter k. Finally, it is concluded that the obtained method gives an effective, efficient, and systematic method. [ABSTRACT FROM AUTHOR]