FFF #63. An Euler Equation

FFF #63. An Euler Equation

Author(s): Ed Barbeau

The College Mathematics Journal, Vol. 24, No. 4 (Sep., 1993), pp. 343-344.

Abstract. In mathematics, there are instances where an individual produces an error when executing a solution to a mathematical inquiry. By addressing some of the common inconsistencies, we are able to evolve from these instances and reduce the probability of committing a similar type of error. As Ed Barbeau clearly depicted in "FFF #63. An Euler Equation," there are times where an individual might assume that a specific equation is a solution to a differential equation. In the case of a second-order homogeneous differential equation, such as Euler’s equation that is expressed as $ax^2y'' + bxy' + cy = 0$, one might consider the equation $y = e^{rx}$ as a trial solution to the second-order homogeneous differential equation $x^2y'' - 5xy' + 9y = 0$. However, when the induvial substitutes this trial solution into $x^2y'' - 5xy' + 9y = 0$, we can conclude that this equation cannot be classified as a solution to the second-order homogenous differential equation. Even though the individual produced an error, we can identify the inconsistency and execute a new trial solution, such as $y = x^r$. If the induvial substitutes this trial solution into $x^2y'' - 5xy' + 9y = 0$ and utilizes methods, such as reduction of order, to solve this second-order homogeneous differential equation, then we obtain the plausible solution $y = c_1x^3 + c_2x^3\ln(x)$. Bryn Brakefield, Stephen F. Austin State University, May 5, 2017.