Mathematician from HSE University–Nizhny Novgorod Solves Equation Considered Unsolvable in Quadratures Since 19th Century

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Mathematician Ivan Remizov from HSE University–Nizhny Novgorod and the Institute for Information Transmission Problems of the Russian Academy of Sciences has made a conceptual breakthrough in the theory of differential equations. He has derived a universal formula for solving problems that had been considered unsolvable in quadratures for more than 190 years. This result fundamentally reshapes one of the oldest areas of mathematics and has potential to have important implications for fundamental physics and economics. The paper has been published in Vladikavkaz Mathematical Journal.

In high school mathematics classes, students learn that to find x in the equation ax²+bx+c=0, it is enough to substitute the coefficients a, b, and c into a ready-made formula for calculating the root via the discriminant. This approach is convenient, fast, and clear. In higher mathematics, however, which is used to describe complex processes, equations of the form ay″+by′+cy=g are employed. These are also second-order equations, but unlike algebraic ones, they are differential equations.

Imagine you are driving a car. If the road is perfectly level and your speed is constant, calculating the travel time is easy. This is a problem with constant coefficients. Now imagine that the road surface is constantly changing, the wind blows with varying strength, and the slope of the mountain beneath the wheels is never the same. Under such conditions, both your speed and travel time depend on multiple, continuously changing factors.

Mathematically, this situation is described by second-order differential equations. In such equations, ordinary numbers are replaced by functions that serve as coefficients—quantities that are themselves constantly changing. And instead of simple squaring, the equation involves taking a second-order derivative, the mathematical analogue of how sharply the car accelerates or slows down.

Such equations are a fundamental tool of science: they describe phenomena ranging from the oscillations of a pendulum and signals in power grids to the motion of planets. Yet this is precisely where researchers reached a dead end. In 1834, the French mathematician Joseph Liouville showed that the solution to such equations cannot be expressed in terms of their coefficients using a standard set of operations—addition, subtraction, multiplication, and division—or elementary functions such as roots, logarithms, sines, cosines, and integrals. Since then, the belief has taken root in the mathematical community that no general formula exists for solving such equations. For more than 190 years, the problem was considered closed and hopelessly unsolvable. The search for a simple formula—analogous to the quadratic formula based on the discriminant—was long ago abandoned in the case of differential equations.

Ivan Remizov, Senior Research Fellow at HSE University and the Institute for Information Transmission Problems of the Russian Academy of Sciences, proposed an elegant solution. Instead of challenging Liouville’s result, he simply expanded the mathematical toolkit. Alongside the standard operations, he added one more element: taking the limit of a sequence. This made it possible to write a formula into which the coefficients a, b, c, and g of the equation ay″+by′+cy=g can be substituted to obtain its solution—the function y. 

The method is based on the theory of Chernoff approximations. Its core idea is to break down a complex, continuously changing process into an infinite number of simple steps. For each step, an approximation is constructed—an elementary fragment that describes the behaviour of the system at a specific point. Individually, these fragments provide only a simplified picture, but as their number approaches infinity, they merge seamlessly into a perfectly accurate solution. The rate at which these approximations converge to the exact solution can be determined using estimates that Ivan Remizov and his colleague Oleg Galkin obtained last year.

In his new paper, Remizov demonstrates that by applying the Laplace transform to these steps—a method that translates a problem from the language of complex change into that of ordinary algebraic calculations—they converge unmistakably toward the final result. Scientists refer to this as a ‘resolvent.’ 

Ivan Remizov

'Imagine that the solution to the equation is a large, intricate picture. It is impossible to take it in all at once. Fortunately, mathematics excels at describing processes that unfold over time. The result is a theorem that allows this process to be “sliced” into many small, simple frames. Then, by applying the Laplace transform, these frames can be assembled into a single, static image—the solution to the complex equation, known as a resolvent. Simply put, rather than trying to guess what the picture looks like, the theorem allows reconstructing it by quickly scrolling through the “film” of its creation,' explains Ivan Remizov, Senior Research Fellow at the International Laboratory of Dynamical Systems and Applications from the Nizhny Novgorod campus of HSE University. 

Second-order differential equations are used not only to model real-world phenomena but also to define new functions that cannot be expressed in any other way. These include, for example, the Mathieu and Hill functions—special functions that are critically important for understanding the motion of satellites in orbit or the behaviour of protons in the Large Hadron Collider.

'The only working definition of such functions is as solutions to specific, complex equations. It’s like not knowing a person’s name and being able to describe them only by what they do—for example, “the person who drives the red bus on Route Five.” You know exactly who you mean, but in practice, it doesn’t help you address them directly by name,' Remizov explains.

The approach proposed by the author makes it possible to express the solutions of equations directly in terms of their coefficients. Thanks to this, special functions can now be defined by explicit formulas—just as the formula y(x) = x² defines the function y. To compute y(x) in this example, one simply multiplies x by itself. While the formulas for Mathieu and Hill functions are more complex, the principle is the same: the value to be found appears on the left side of the equation, and the explicit steps to calculate it are on the right.

At the same time, Ivan Remizov’s work serves as a bridge between mathematics and modern physics. For the first time, he presented a solution to an ordinary differential equation as a formula reminiscent of the famous integrals introduced by Nobel laureate Richard Feynman, which describe the motion of quantum particles. What once applied only to quantum mechanics can now be applied to classical problems as well.

‘However, we should not be overoptimistic. This is a new, but not the very first method of solving such equations. There are Peano-Baker series, product integral, probabilistic methods—for example, Feynman-Kac formulas. Also, for the special functions there are many known representations, including power series. The main advantage of the approach proposed is the simplicity of the result. Everyone who has a bachelor’s degree in mathematics, physics or engineering can easily understand all the operations that are used in this new formula for the solution,' Ivan Remizov comments.