# Transforming some Differential Equations into Algebraic Differential Equations

- Juan Rafael Sendra (Universidad de Alcalá, Alcalá de Henares, Spain)

### Abstract

In the initial courses of calculus, when dealing with the computation of primitives, it is shown how certain integrals, whose integrands involve radicals of polynomials, can be transformed by a change of variable into integrals whose integrand are rational functions; and, hence, handleable algebraically. If the indefinite integral is seen as a differential equation, the natural question is whether we could develop a similar strategy to transform differential equations, whose coefficients involve radicals of polynomials, into differential equations whose coefficients are now rational functions. Illustrating examples of this are \[4(y')^3\sqrt{x+1}^{\,3} -(x+1)yy' + y^2 = 0,\] or \[ \left(-\sqrt{x_2}\,\dfrac{\partial y(\overline{x})} {\partial x_3} +2\,\dfrac{\partial y(\overline{x})} {\partial x_1} \right)\sqrt{x_1+\sqrt{x_2}} + 2\sqrt{x_2}\,\dfrac{\partial y(\overline{x})} {\partial x_2} - y(\overline{x})^2 - \dfrac{\partial y(\overline{x})} {\partial x_1}=0,\] or even $$\sqrt{x}\,y''-\sqrt{y^{3}} = 0.$$

In this talk we will recall some basic facts on radical varieties, and we will present an algorithm to transform, if possible, a given differential equation, with radical function coefficients, into one with rational function coefficients by means of a rational change of variables so that solutions correspond.