Introduction to differential equations
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This free OpenLearn course, Introduction to differential equations, is an extract from the Open University module MST124 Essential mathematics 1, covering a wide range of topics from different areas of mathematics. The practical application to problems provides a firm foundation for further studies in mathematics and other mathematically rich subjects such as physics and engineering. Topics covered include mathematical typesetting, number theory, conics, statics, geometric transformations, calculus, differential equations, mathematical language and proof, dynamics, eigenvalues and eigenvectors and combinatorics. It also helps develop the abilities to study mathematics independently, to solve mathematical problems and to communicate mathematics.. The module builds on mathematical ideas introduced in
Introduction to differential equations consists of material from MST125 Unit 8, Differential equations and has six sections in total. Sections 5 and 6 are considerably shorter than the other four sections. You should set aside about four hours to study each of the first four sections and about two hours for the remaining two sections; the whole extract should take about 18 hours to study. The extract is a small part (around 8%) of a large module that is studied over eight months, and so can give only an approximate indication of the level and content of the full module.
The extract, which contains an introduction to differential equations, is relatively self-contained and should be reasonably easy to understand for someone who has not studied any of the previous texts in MST125. A few techniques and definitions taught in earlier units in MST125 are present in the extract without explanation, therefore a fluency with algebra and basic calculus is essential for this extract.
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 MST125 software or to video material (although please note that the PDF may contain references to other parts of MST125). In this extract, some illustrations have also been removed due to copyright restrictions.
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Differential equations are any equations that include derivatives. They arise in many situations in mathematics, physics, chemistry, engineering, biology, economics and finance. Three types of first-order differential equations are considered.
Section 1 introduces you to equations that can be solved by direct integration.
Section 2 introduces the method of separation of variables for solving differential equations.
Section 3 looks at applications of differential equations for solving real world problems including variations in the size of a population over time and radioactive decay.
Section 4 introduces you to the integrating factor method for solving linear differential equations.
The final two sections summarise and revise the methods introduced in the previous sections and describe various other approaches to finding solutions of first-order differential equations and to understanding the behaviour of the solutions.