### A Method for Solving the Special Type of Cauchy-Euler Differential Equations and its Algorithms in MATLAB

#### Abstract

In many applications of sciences, for solve many them, often appear equations of type N-Order Linear differential equations, where the number of them is Cauchy-Euler differential equations (also known as the Euler differential equation). i.e. Cauchy-Euler differential equations often appear in analysis of computer algorithms, notably in analysis of quicksort and search trees; a number of physics and engineering applications, such as when solving Laplace's equation in polar coordinates; and many other sciences.

In this paper, we use variable change method for solving the special type of N-Order Cauchy-Euler differential equations. We applying a variable change in the equation, and then obtain the conditions, where if we have an equation that applies to these conditions, a simple analytical solution for the equation can be obtained. Because in this method, an analytical solution is obtained, therefore, it is not necessary to use numerical methods to solve the problem.

In this paper, we use variable change method for solving the special type of N-Order Cauchy-Euler differential equations. We applying a variable change in the equation, and then obtain the conditions, where if we have an equation that applies to these conditions, a simple analytical solution for the equation can be obtained. Because in this method, an analytical solution is obtained, therefore, it is not necessary to use numerical methods to solve the problem.

#### Keywords

Cauchy-Euler differential equation; Euler differential equation; ordinary differential equation; linear differential equations; Equidimensional equations; Wronskian; MATLAB

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