Differential Equations with Applications to Vibrations

**Ines Radošević Medvidović**, *University of Rijeka (Croatia)*

**Melita Štefan Trubić**, *University of Rijeka (Croatia)*

Abstract

New findings in science and new information that araise from technology are necessary to be reflected in teaching. So, modern education is necessarily education of modern science. Modern teaching of mathematics and physics is not only the interest of themselves, but also the interest of different techniques that serve them. Lack of student motivation during the teaching process and passing on knowledge of mathematics does not necessarily have to be "boring" as it can avoided by selecting the most convenient teaching methods and applications in practice. In particular, teaching of mathematics at the technical college should be focused on teaching and acquiring knowledge of engineering mathematics, i.e., applied mathematics for engineers, physicists, mathematicians and computer scientists, as well as on other areas of scientific disciplines. Students need a solid knowledge of the basic principles, methods and results from mathematical areas, and everything that requires engineering mathematics, as well as the expertise in all three phases of problem solving. The first step is modeling, i.e. putting a physical or other problem into a mathematical form. This can be a differential equation, as shown here. Secondly, solving a model by applying a suitable mathematical method, which often requires the need to work on your computer, and finally, interpretation of mathematical results in the physical sense, to see and understand what it actually means and implies. This paper presents a teaching lesson in mathematics at the technical college: "Linear ordinary differential equations (ODEs) of the second order". After defining and discussing the second order homogeneous and nonhomogeneous liner ODEs, we are going to take a closer look at practical meaning of mechanical vibration, where linear ODEs with constant coefficients have great and important applications. In our particular case, the system with one degree of freedom, i.e. the mechanical system of mass hanging from the spring. This paper shows some examples to illustrate harmonic damped oscillations and corresponding behavior of the system in case of critical damping, over-damping and under-damping, and undamped and damped forced oscillations using Matlab or Mathematica.