A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients

Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the new analytical method. The new method was derived from the concept of exactness in higher-order ODEs. Specifically, second- and third-order ODEs with variable coefficients are exact if there exist first- and second-order linear ODEs whose derivatives correspond to the given equations, respectively. In this new analytic method, an integrating factor function formula for second-order ODEs has been carefully formulated and derived, making every second-order ODE with variable coefficients reducible to its lower-order form, specifically first-order ODEs. To ensure the accuracy of the new method, two well-known classes of second-order linear ODEs, namely the Whittaker second-order linear ODE and the Modified Bessel equation, were applied. The results demonstrated that the new analytic method effectively solves these equations, producing exact analytical solutions. To validate the effectiveness and efficiency of the new analytic method, a comparative analysis was conducted using illustrative examples, followed by graphical representations of the solution results.