Abstract
In this article, we will explore the applications of linear ordinary differential equations (linear ODEs) in Physics and other branches of mathematics, and dig into the matrix method for solving linear ODEs. Although linear ODEs have a comparatively easy form, they are effective in solving certain physical and geometrical problems. We will begin by introducing fundamental knowledge in Linear Algebra and proving the existence and uniqueness of solution for ODEs. Then, we will concentrate on finding the solutions for ODEs and introducing the matrix method for solving linear ODEs. Eventually, we will apply the conclusions we’ve gathered from the previous parts into solving problems concerning Physics and differential curves. The matrix method is of great importance in doing higher dimensional computations, as it allows multiple variables to be calculated at the same time, thus reducing the complexity.
Highlights
Ordinary Differential Equations (ODEs) has a broad range of applications in other subjects, such as Physics and Differential Geometry
We will explore the applications of linear ordinary differential equations in Physics and other branches of mathematics, and dig into the matrix method for solving linear ODEs
We will concentrate on finding the solutions for ODEs and introducing the matrix method for solving linear ODEs
Summary
Ordinary Differential Equations (ODEs) has a broad range of applications in other subjects, such as Physics and Differential Geometry. We will explore the applications of the matrix method in linear ordinary differential equations (linear ODEs) in Physics and other branches of Mathematics. Our ultimate aim is to be able to solve any given linear ordinary equations using constant matrices as parameters, and apply such a computational method into solving the evolution function of the movement of a free electron in a constant three-dimensional electric field and proving the Frenet formula in calculating the torsion of a 3-D curve (Fage M.K., 1974) [4]. Example 1 (Radioactive chains of decay) The differential equation for the number N of radioactive Nucleus is dN = −kN (1.1). The third example is, difficult to solve because the magnetic field B is a three-dimensional matrix instead of a constant. Our goal of this article is to be able to solve the differential equation of the type of example 3
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