Firstly, this paper introduces the definition, some basic concepts and development history of ordinary differential equations. Then, on the basis of summarizing some common methods for solving ordinary differential equations, the concept of linear differential equations is introduced. Finally, the solution of two-dimensional non-homogeneous linear differential equation with variable coefficients is studied emphatically, and the special solution formula of non-homogeneous linear differential equation with variable coefficients is studied by using the concept of matrix and combining linear algebra and differential equations. In addition, the specific equation is given and the solved expression is tested.

Keywords: differential, equation, linearity, special solution

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