Transforming some Differential Equations into Algebraic Differential Equations

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.