D'Alembert's formula is only suitable for a few definite solution problems, and its solution idea is to find the general solution from the universal equation itself, without considering any additional conditions. Generally, the general solution will contain an integral constant, and then the integral constant will be determined by using additional conditions. This process is similar to solving ordinary differential equations.

The separation of variables method transforms partial differential equations into several ordinary differential equations by using boundary conditions, transforms boundary conditions into additional conditions to form eigenvalue problems, and then uses initial conditions to find corresponding coefficients.

The method of separating variables is to decompose a partial differential equation into two or more ordinary differential equations with only one variable. By separating the terms containing various variables in the equation, the original equation is divided into several simpler ordinary differential equations with only one independent variable.

Using the principle of linear superposition, the non-homogeneous equation is decomposed into several homogeneous or easy-to-solve equations.

Extended data:

Mathematically, the method of separating variables is a method to analyze ordinary differential equations or partial differential equations. Using this method, we can rearrange the equation by algebra, so that part of the equation contains only one variable, while the rest has nothing to do with this variable. In this way, the values of the isolated two parts are equal to constants, and the algebraic sum of the values of the two parts is equal to zero.

The general solution of each equation is obtained by using clever methods such as high number knowledge and series solution. Finally, these common solutions are "assembled". Separation of variables is a common method to solve the initial-boundary value problem of wave equation.

Baidu Encyclopedia-Variable Separation Method