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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.

## Units

### 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.

Lesson

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.

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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