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Differential Equations (Metric)

Class Details

Tom Baker

Learn, or revise, solving first order differential equations by various methods, solving second order differential equations with constant coefficients, and classifying the critical points of systems.

First Order

Solving differential equations of the form \(dy/dx=f(x,\,y)\). Part of this topic is found on A-level courses, and the rest is Further Maths.

Solving differential equations of the form \(dy/dx = f(x)\,g(y)\), by separating the variables.

Lesson Not yet graded

Solving differential equations that can be written in the form \(d/dx(f(x,\,y))=0\).

Lesson Not yet graded

Second Order

Solving differential equations of the form \(a\,d^2y/dx^2+b\,dy/dx+c\,y=f(x)\). This is a Further Maths topic; courses that require Further Maths may assume it.

Solving differential equations of the form \(a\,d^2y/dx^2+b\,dy/dx+c\,y=0\); oscillatory and exponential solutions.

Lesson

Solving differential equations of the form \(a\,d^2y/dx^2+b\,dy/dx+c\,y=f(x)\) using a complementary function and a particular integral.

Lesson

Strong, critical and weak damping; resonance.

Lesson

Qualitative Methods

Classifying the critical points of systems of the form \(dx/dt = f_1(x,\,y)\), \(dy/dt = f_2(x,\,y)\). This is part of the First Year content of some Science and Engineering courses at Imperial.

Classifying the critical points of systems of the form \(dx/dt = a_{11}\,x+a_{12}\,y\), \(dy/dt = a_{21}\,x+a_{22}\,y\), based on the eigenvalues of the coefficient matrix.

Lesson

Classifying the critical points of systems of the form \(dx/dt = f_1(x,\,y)\), \(dy/dt = f_2(x,\,y)\), where \(f_1\) and \(f_2\) are nonlinear, using local linearisation or by considering the Jacobian.

Lesson
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