Linear differential equation

A linear differential equation is a mathematical rule used to find an unknown function. This rule relates the function to its successive derivatives, which measure how the function changes. The equation is called "linear" because the unknown function and its derivatives appear only to the first power. They are not squared, cubed, or tucked inside other functions. These equations can be ordinary differential equations (ODEs) if they involve one variable. They are called partial differential equations (PDEs) if the unknown function depends on several variables. This distinction is important because PDEs use partial derivatives to track changes across multiple dimensions.

To understand the structure of these equations, we must identify their specific parts. The highest order of derivation present in the equation defines its order. For example, an equation containing a second derivative is a second-order equation. We also look at the coefficients, which are the functions or numbers multiplying the derivatives. If these coefficients are just fixed numbers, we call them constant coefficients. There is also a term called the constant term, which does not depend on the unknown function. If this term is zero, the entire equation is called homogeneous. If the term is not zero, the equation is non-homogeneous.

Mathematicians often use a tool called a linear differential operator to simplify these equations. An operator is a mapping that takes a function and turns it into its derivative. A linear operator is a combination of these basic mappings using differentiable functions as coefficients. We can write a complex equation very compactly by using this operator notation. The solutions to a homogeneous linear differential equation are quite special. They form a vector space, which is a mathematical collection of functions. For an ordinary differential equation of order $n$, this vector space has exactly $n$ dimensions.

History shows that the study of these equations has deep roots. The mathematician Leonhard Euler made massive contributions to this field. He studied equations with constant coefficients and introduced the exponential function, $e^x$. This function is unique because it is the only solution to a specific equation where the starting value is one. Euler's work allowed mathematicians to solve homogeneous equations more easily. By searching for solutions in the form of $e^{rx}$, they could turn calculus problems into algebra problems. This method involves finding the roots of a characteristic polynomial to find the basis of the solution space.

Solving these equations depends heavily on the type of equation being studied. For equations of the first order with non-constant coefficients, solutions can often be found by quadrature. This means the solution can be expressed using integrals. However, equations of the second order or higher with non-constant coefficients are much harder. They cannot always be solved by quadrature. For second-order equations, a tool called Kovacic's algorithm can help decide if an integral-based solution exists. For second-order equations with constant coefficients, the solution depends on a discriminant. This can result in distinct real roots, a single double root, or complex conjugate roots.

There are several advanced methods used to solve non-homogeneous equations. One approach is the method of undetermined coefficients, which works well if the non-homogeneous part is a combination of exponential or sinusoidal functions. Another method is the annihilator method, which is used when the non-homogeneous part is a holonomic function. Perhaps the most general approach is the variation of constants. This method treats the constants in a homogeneous solution as unknown functions instead. By solving a system of linear equations, mathematicians can find the specific functions needed to satisfy the non-homogeneous rule. This allows for a complete solution that combines the homogeneous and particular parts.

Linear differential equations are connected to a vast family of functions known as holonomic functions. These functions are the solutions to homogeneous linear differential equations that have polynomial coefficients. This class is very stable, meaning it stays intact under addition, multiplication, differentiation, and integration. It includes many familiar mathematical tools, such as sine, cosine, logarithms, and exponential functions. It also includes more specialized tools like Bessel functions and hypergeometric functions. Because of their predictable nature, we can use algorithms to calculate their limits, antiderivatives, and numerical values with very precise error bounds.