<p>This work investigates the Cauchy problem for a high-order ordinary differential equation involving the Bessel operator and a spectral parameter. Due to the lack of appropriate tools, this problem has not been previously studied. The primary aim of the paper is to introduce one such tool, namely the generalized Erdélyi–Kober fractional operator, which possesses the property of a transmutation operator. By applying this operator, the considered problem is reduced to an equation without degeneration and without a lower-order term. An explicit formula for the solution of the problem is constructed. Another aim is to demonstrate the effectiveness of the proposed method, which allows for obtaining an exact solution to the formulated problem. Despite the advancement of modern computational tools, constructing exact solutions for boundary value problems for ordinary differential equations remains an important and relevant task. Such solutions provide deeper insight into the qualitative properties of the processes and phenomena being described, the properties of mathematical models, and may also serve as benchmark examples for asymptotic, approximate, and numerical methods.</p>