Second-order differential equations
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This free OpenLearn course, Second-order differential equations is an extract from the Open University course, a second level course that provides a background in mathematical methods for students studying physical sciences, engineering, mathematics and economics. The course is designed to teach ‘pencil and paper’ techniques that are at the core of applying mathematics. Students are expected to be familiar with standard mathematical notation and with elementary functions such as sines, cosines, logarithms and exponentials. Familiarity with algebra, calculus and complex numbers is also assumed.
Second-order differential equations consists of material from MST224 Unit 3, Second-order differential equations and has five sections in total. You should set aside between three to four hours to study each of the sections; the whole extract should take about 16 hours to study. The extract is a small part (around 8%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.
This extract is the core calculus unit from MST224, covering second-order differential equations and should provide a fair idea of what is in the full course – lots more calculus. It takes an approach of illustrating all of the methods with concrete physical applications; this is typical of MST224. It is relatively self-contained and should be reasonably easy to understand for someone who has not studied any of the previous texts in the course. However, a few techniques and definitions taught in earlier units in MST224 are present in the extract without explanation.
Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the MST224 software or to video material (although please note that the PDF may contain references to other parts of MST224). In this extract, some illustrations have also been removed due to copyright restrictions.
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Section 1 introduces some basic principles and terminology.
Sections 2 and 3 give methods for finding the general solutions to one broad class of differential equations, that is, linear constant-coefficient second-order differential equations. Section 2 covers homogeneous equations and Section 3 covers inhomogeneous equations.
Section 4 explains how extra information can be used to help to select particular solutions that are appropriate in given situations.
Section 5 looks at the phenomenon of resonance to gain insights into the solutions of inhomogeneous linear constant-coefficient second-order differential equations.