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Introductory Calculus: Oxford Mathematics 1st Year Student Lecture

Oxford Mathematics・22 minutes read

The course covers differential equations, integration, and practical information, with lectures by Dan Ciubotaru and notes by Cath Wilkins available online. Topics include ODEs, PDEs, integration techniques, and examples in mechanics and engineering, with a recommended reading list and focus on interactions with other math courses.

Insights

  • The course covers a wide range of topics, including practical information, examples of differential equations, integration techniques, and a recommended reading list. Lectures are delivered by Dan Ciubotaru, with online notes available by Cath Wilkins, and students meet twice a week at 10 am for a total of 16 lectures.
  • The course delves into practical applications of differential equations in mechanics, engineering, and radioactive decay, showcasing real-world relevance. Integration techniques like integration by parts are demonstrated with examples, emphasizing the importance of recursive formulas and careful consideration when dividing by functions involving variables like y in differential equations.

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Recent questions

  • What topics are covered in the course?

    Various topics like differential equations, integration, and calculus.

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Summary

00:00

"Differential Equations and Integration Course Overview"

  • Course covers practical information, syllabus, examples of differential equations, and integration.
  • Lectures are 16 in total, notes online by Cath Wilkins, lecturer is Dan Ciubotaru, meeting twice a week at 10am.
  • Eight problem sheets, first two online, remaining covered in four tutorials in college.
  • Reading list online, recommended book is Mary Boas's "Mathematical Methods in Physical Sciences."
  • Syllabus includes differential equations (ODEs and PDEs), integration, line and double integrals, calculus of functions in two variables.
  • Interaction with other courses like multivariable calculus, dynamics, PDEs, and analysis two.
  • Differential equations examples in mechanics (Newton's second law), engineering (RLC circuit), and radioactive decay.
  • Integration techniques like integration by parts demonstrated with examples.
  • Example integrals solved: x^2 sin(x) and 2x - 1 Ln(x^2+1)dx.
  • Long division used in integration example to solve x^3 - x^2 over x^2 + 1.

30:33

Integrals, Differential Equations, and Recursive Formulas

  • The integral J is computed as the integral of (x - 1 + (-x + 1))/(x^2 + 1) dx, resulting in x^2 - 1/2 x^2 - 6.
  • To integrate the term x/(x^2 + 1) dx, it is split into x/(x^2 + 1) + dx/(x^2 + 1).
  • The last term of the integral is recognized as arctan(x) or tan inverse of x.
  • Substitution is done by letting u = x^2 + 1, leading to the final integral J as 1/2 x^2 - x - 1/2 Ln(x^2 + 1) + tan inverse of x + constant.
  • Integration by parts may not simplify functions f and g, but repeating the process can lead back to the original function.
  • An example is given with the integral e^x * sin(x) dx, which is solved through multiple integration by parts steps.
  • Recursive formulas are used for more complex integrals, demonstrated with an example involving cos^n(x) dx.
  • The general solution to a separable differential equation is found by separating variables and integrating each side.
  • An example is provided with x * y^2 - 1 + y * x^2 - 1 dy/dx = 0, leading to an implicit equation in y and x with a constant C.
  • Care must be taken when dividing by functions involving y, as solutions may arise when the function equals zero.
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