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Differential Equations (Metric)

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.

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

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.

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.

Strong, critical and weak damping; resonance.

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.

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.