In this blog post, we’ll examine basic methods for solving differential equations and explore how mathematics can be applied to art authentication through a case study in which these methods were used to identify a counterfeit painting.
For most people, differential equations are an unfamiliar topic, and it’s easy to think of them as extremely difficult math problems. In fact, many differential equations are tricky to solve, and there are still problems in this field that even experts cannot solve. However, even relatively simple differential equations can yield a wealth of information when their solution methods are applied to other fields. Differential equations allow us to express various laws of nature mathematically, and solving these equations helps us understand the phenomena that occur in nature. It is no exaggeration to say that this has led to countless advancements not only in the natural sciences but also in engineering. In this article, we will examine the solution to a simple form of a differential equation that anyone can understand relatively easily, and explore how differential equations were used to identify the forged artworks of the 20th-century painter Han van Meegeren.
A differential equation is an equation consisting of an unknown function and its derivative. While a general equation consists of unknown variables and aims to find the values of those variables, a differential equation aims to find the unknown function. Differential equations can be classified in various ways depending on their form. A differential equation with a single independent variable is called an ordinary differential equation, while one with multiple independent variables is called a partial differential equation. Furthermore, the order of a differential equation is determined by the highest-order derivative it contains. For example, if the highest-order derivative is an nth-order derivative, the equation is called an nth-order differential equation. The number of differential equations we can solve by hand is limited, and even a slight increase in complexity makes solving them very difficult. Today, with the help of computers, we can solve even more complex forms of differential equations; however, not all differential equations can be solved perfectly, and they remain problems for mathematicians to study.
Let’s first look at the solution to one of the simplest forms of differential equations: y′ + y = 1. Here, y is a function of t, and y′ is the derivative of y, which can also be expressed as dy/dt. Since this differential equation involves a linear combination of the function y and its derivative y′, the derivative is of the first order, and there is only one independent variable, t, it is a first-order ordinary differential equation. To solve it, we use the fact that the derivative of the product of two functions f and g is (fg)′ = f′g + fg′. Multiplying both sides of the differential equation by eᵗ yields eᵗy′ + eᵗy = eᵗ. However, the derivative of eᵗ is eᵗ, and by the product rule, we see that (eᵗy)′ = eᵗy′ + eᵗy. Therefore, rewriting the left-hand side, our differential equation becomes (eᵗy)′ = eᵗ, and we can find the solution by integrating both sides. In general, for a differential equation of the form above, suppose the initial value y(t₀) = y₀ is given at t = t₀. If a solution to the first-order ordinary differential equation exists and appropriate conditions are satisfied, then a single initial value yields a unique solution.
Therefore, by integrating both sides from t₀ to t, we obtain eᵗy − eᵗ⁰y₀ = ∫ₜ₀ᵗ eˢ ds = eᵗ − eᵗ⁰. Rearranging this for y, we can confirm that the unknown function we seek is y = 1 + (y₀ − 1)e⁻⁽ᵗ⁻ᵗ⁰⁾ .
Now, let’s find the solution to a more general form of a differential equation, y′(t) + a(t)y(t) = b(t), using the same method as in the previous example. While in the previous example a(t) and b(t) were both equal to 1, in this case a(t) and b(t) are functions of t. Although it may seem complicated due to the inclusion of general variables, the solution method remains the same. In the previous example, we multiplied both sides by e^t to express the entire left-hand side in differential form. This time, let’s multiply both sides by μ(t) = exp(∫a(t)dt) instead of e^t. Here, μ(t) is called the integrating factor and plays a key role in solving the differential equation. We then obtain a differential equation of the form μ(t)y′(t) + μ(t)a(t)y(t) = μ(t)b(t). Since μ′(t) = a(t)exp(∫a(t)dt) = a(t)μ(t), the left-hand side can be rewritten in a single derivative form as μ(t)y′(t) + μ′(t)y(t) = (μ(t)y(t))′. Consequently, the differential equation takes the form (μ(t)y(t))′ = μ(t)b(t). By integrating both sides and rearranging for y, we can find the solution to the differential equation. Now, let’s examine an interesting case in which this differential equation was used to identify a forged work of art.
Han van Meijeren was a painter famous for forging works by the Dutch artist Johannes Vermeer. After World War II ended, van Meijeren came under investigation on suspicion of selling paintings purported to be by Vermeer to Hermann Göring, a high-ranking German official. Because he had sold valuable Dutch artworks to Nazi Germany, Meijer was on the verge of being charged with collaboration with war criminals. However, during the investigation, Meijer claimed that several works—including the one he had sold to Göring—were forgeries he had created himself. This confession threw the art world into turmoil and sparked a debate over whether the various works Meijer had traded were genuine or forged. Subsequently, various studies were conducted to scientifically determine the authenticity of artworks, including a method that used differential equations to model the decay characteristics of radioactive elements in order to estimate the age of pigments used in paintings and assess their authenticity. In particular, during the 1960s, a method utilizing the radioactive decay of lead-210 to estimate the production date of white pigments was applied to art authentication. This method was used to analyze paintings subject to authenticity disputes, including Meyheren’s forgeries.
White lead, a white pigment used in paintings, contains lead, and the decay characteristics of its associated radioactive isotopes—lead-210 (²¹⁰Pb) and radium-226 (²²⁶Ra)—can be utilized. Radioactive elements decay over time to form other elements, and the decay rates of each element differ. Generally, the decay rate of a specific radioactive element is proportional to the amount of that element currently present. Therefore, if N(t) represents the number of a specific radioactive element at time t, its rate of change can be expressed by the differential equation dN(t)/dt = −λN(t). Here, λ is the decay constant of the element in question. Since radioactive elements decay in a chain reaction to produce new elements, the decay rate of one element and the decay rate of the resulting element influence each other. In the case of lead-210 contained in the white lead used in paintings, this relationship can be used to explain how its quantity changes over time. Radium-226 forms a radioactive series that leads to lead-210 during its decay process, and lead-210 subsequently decays. Initially, the rate at which lead-210 is produced from radium-226 and the rate at which lead-210 decays may not be in equilibrium; however, as time passes, the relationship between the two processes changes, and under certain conditions, it approaches radioactive equilibrium.
Let y denote the amount of lead-210 in the painting and r denote the rate at which lead-210 is produced from radium-226; then, the change in lead-210 can be expressed by the differential equation y′ = −λy + r. Here, y represents the amount of lead-210, λ is the decay constant of lead-210, and r is the rate at which lead-210 is produced as a result of the decay of radium-226. Since radium-226 has a very long half-life of approximately 1,600 years, the decay rate r can be assumed to be constant when dealing with time scales of several hundred years. Since this differential equation has the same form as the first-order linear differential equation examined earlier, it can be solved using the same method. As a result, we obtain the solution y = r/λ(1 − e⁻λ⁽ᵗ⁻ᵗ⁰⁾⁾) + y₀e⁻λ⁽ᵗ⁻ᵗ⁰⁾. By measuring the current decay rate of lead-210, the decay rate of radium-226, and other relevant parameters from this equation, we can analyze the state of radioactive material in a painting.
While r (the decay rate of radium-226), λ (the decay constant of lead-210), and the current decay rate of lead-210, λy, are values that can be determined through measurement, the amount of lead-210, y₀, present at the time the pigment was manufactured is difficult to determine directly; therefore, it is difficult to calculate the exact time t − t₀ when the artwork was created using only the above equation. However, assuming the artwork is authentic and was created in the 17th century—meaning it was produced approximately 300 years ago—we can substitute 300 into t − t₀ to calculate the decay rate of lead-210 at that time. Rearranging this into an equation yields λy₀ = λy(t)e³⁰⁰λ − r(e³⁰⁰λ − 1). Since the half-life of lead-210 is approximately 22 years, λ can be set to ln2/22; using this, we can calculate the expected initial decay rate of lead-210 in white lead manufactured 300 years ago. Based on the relationship between radium-226 and lead-210 measured in various historical ores, this value should not be excessively high in old white lead presumed to be genuine. Conversely, in forgeries made with modern white lead, lead-210 has not reached sufficient radioactive equilibrium with radium-226; therefore, the current decay rate of lead-210 may appear abnormally high compared to that of radium-226. By exploiting this difference, it is possible to determine whether the white lead used in a painting was produced in the past or relatively recently. The study concluded that for white lead approximately 300 years old—which could be considered authentic—it was difficult to accept a value of λy₀ significantly exceeding 30,000, and this criterion was used to determine whether a work was a forgery.
This method was applied to determine the authenticity of ‘Disciples at Emmaus’, a work previously attributed to Vermeer. Analysis of the white lead in this painting revealed that the decay rate of polonium-210—measured as a substitute for the lead-210 decay rate—was 8.5, while that of radium-226 was 0.8. Since lead-210 eventually reaches a state of radioactive equilibrium with its decay product, polonium-210, over sufficient time, the decay rate of polonium-210—which is relatively easy to measure—can be used as an approximation for the decay rate of lead-210. Substituting this value into the equation above yields a value of λy₀—corresponding to the decay rate of lead-210 at the time the white lead was manufactured—of approximately 98,050. This value falls far outside the range expected for authentic white lead that is approximately 300 years old. Therefore, the white lead used in this work is incompatible with the pigments used in authentic 17th-century pieces, supporting the conclusion that it is a forgery created using modern pigments. Such analyses have been applied not only to ‘Disciples at Emmaus’ but also to several other Vermeer works subject to authenticity disputes, and in some cases, the results have confirmed that they are forgeries.
Differential equations allow us to mathematically describe phenomena occurring in nature and analyze their changes. As seen in the example above, the solutions to differential equations can be applied even in the field of art—which may seem entirely unrelated to mathematics—to analyze changes in the radioactive properties of pigments used in paintings and determine whether a work is a forgery. The scope of applications for differential equations does not end there. Differential equations are applied in countless fields related to human life, ranging from analyzing the motion of an object falling from a height to predicting the motion of celestial bodies and modeling phenomena such as population changes. Efforts to solve differential equations more accurately and quickly and to apply them to various fields continue, and as mathematical methods utilizing differential equations advance, they will contribute significantly to the development of various fields, including the natural sciences and engineering.
