A linear differential equation is a mathematical equation that involves a function and its derivatives. It can be written in the form:
aₙ(x) · yⁿ(x) + aₙ₋₁(x) · yⁿ⁻¹(x) + ... + a₁(x) · y'(x) + a₀(x) · y(x) = g(x)
where y(x) is the unknown function, yⁿ(x) denotes the nth derivative of y(x) with respect to x, and aₙ(x), aₙ₋₁(x), ..., a₁(x), a₀(x), and g(x) are given functions of x.
The coefficients aₙ(x), aₙ₋₁(x), ..., a₁(x), and a₀(x) are called the coefficients of the differential equation, and they are allowed to depend on the independent variable x. The function g(x) is called the forcing function or the inhomogeneous term.
The linearity of the equation implies that the unknown function y(x) and its derivatives appear only with a power of 1, and there are no products or powers of y(x) or its derivatives. Furthermore, the coefficients aₙ(x), aₙ₋₁(x), ..., a₁(x), and a₀(x) can be any functions of x, as long as they are not functions of y(x) or its derivatives.
Solving a linear differential equation typically involves finding the function y(x) that satisfies the equation for all values of x. This can be achieved using various methods, such as separation of variables, integrating factors, or applying specific techniques for solving higher-order linear differential equations.
