# Abstract for the talk on 09.06.2022 (14:00 h)

**Seminar on Nonlinear Algebra**

*Rafael Sendra*(University of Alcalá)

**Transforming some Differential Equations into Algebraic Differential Equations**

09.06.2022, 14:00 h, MPI für Mathematik in den Naturwissenschaften Leipzig, E1 05 (Leibniz-Saal)

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.

Physical attendance in seminars is restricted to 25 participants.

If you want to attend this talk, you need to register in advance. Registration will be open starting seven days before the seminar and registrations will be accepted on first come, first served basis. External participants from Leipzig University need to fill in their name, address, email, and phone number.

Please also check our general Corona rules page. Participants must wear a face mask, and they are encouraged to use the corona rapid tests made available by the institute.