#### Abstract

In this section, we consider ordinary differential equations of first and second order. Topics cover major types of equations such as first order separable equations, linear equations and second order linear equations. The Laplace transform and its application for solving differential is also considered. Examples with detailed solving are presented through the topics to provide guidance on how to approach and solve various problems. Some applications of differential equations are considered in the relation to maritime problems.

#### Aim

To gain understanding what differential equations are and their importance for applications; to acquire skills in solving basic types of first and second order ordinary differential equations and understanding the concepts behind these solutions.

#### Learning outcomes

- Determine the type of a differential equation
- Solve first‐order differential equation using the appropriate solution method
- Solve second‐order linear differential equations with constant coefficients
- Solve linear differential equations using Laplace transform.

#### Previous knowledge of mathematics

*algebra, logarithmic functions and their properties, complex numbers, differentiation and ability to find derivatives, methods of integration, improper integrals, ability to solve systems of linear equations*

#### Relatedness with solving problems in the maritime field

Differential equations are a model of the world. They are widely used for mathematical modeling of processes and phenomena in a variety of fields of science and technology: in mechanics, physics, chemistry, biology, economics and also in modeling many maritime problems. For example, a vessel’s motion, rolling, pitching and heaving motions are described by differential equations. Differential equations are used to describe cooling processes, vibrations, deformation of beams and plates which are important in ship design and in load calculations for cargo transport. Processes in electrical circuits of electric systems of vessels and electro-automatics are also described by differential equations. There are also many marine ecological issues where differential equations are useful, for example, to describe many chemical reactions, to describe the propagation and extinction of various bacteria and microorganisms in the sea, to describe propagation and extinction of fish population that is important for fish catch control.

#### Contents

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