Week-by-week lecture summaries for Akshay Venkatesh

See also Maksim Maydanskiy's summaries for his lectures.

• Weeks 1

  (This isn't broken down by lecture.)

  Basics: what is a differential equation? Classification: first-order versus higher order, linear versus nonlinear, constant coefficients versus general coefficients.

  Some examples of what can be modelled by first-order linear differential equations: population growth, population growth with immigration, bank account with (possibly variable) interest, drug concentration in body assuming constant removal rate, temperature of an object with external cyclic heating/cooling. In each case there is a quantity which grows/decreases in proportion to itself, together with an externally imposed inflow/outflow.

  A little bit about how to solve numerically. This is not part of the syllabus and was just discussed for background knowledge.

  Solving first order linear equations: firstly constant-coefficients, and then the general case via integrating factors. Also "by common sense" in the case of the bank account with variable interest. Solution in the case of the object with cyclic heating/cooling: temperature cycle lags the heating/cooling cycle.

  Week 2

  First order equations which aren't nonlinear. Examples: the logistic model of population growth, falling object with air resistance quadratic in the velocity. Some discussion of the extent to which these are realistic models.

  Separation of variables and many examples of how to use it. Exact solution to logistic model and falling object. Example of an object falling towards earth from space: converting a second order equation to a first-order one, and exact solution of the resulting equation.

  Exact solution to the pendulum equation. Conserved quantities and exact equations -- a little bit.

  Week 3

  Sorry! I didn't record this week.

  Week 4

  Systems of linear differential equations: introduction and examples. Any second order equation can be made into a system of two differential equations. The double pendulum.

  Linear systems can be written in matrix form. When written like this, many techniques from the case of ordinary first order equations carry over, although one might have to use "matrix exponential". Brief discussion of matrix exponential.

  Solving the simple differential equation x'=2x+y, y' = x+2y by means of the substituion x=u-v, y=u+v. This substitution can be understood in terms of eigenvalues of the corresponding matrix.

  Review of eigenvalues and eigenvectors. Every eigenvector of A gives rise to a solution of the system [x', y'] = A [x,y]. The general solution of [x', y'] = A [x,y] is a combination of such "eigenvector solutions." Proof of this by generalizing the u,v substitution discussed above.

  Examples and graphs in the x,y plane. Solutions increase/decrease along the eigenvector directions. Sources and sinks and saddles, according to sign of eigenvalues.

  Week 5

  Quick review of sources/sinks/saddles. Some examples where one eigenvalue is equal to zero.

  Review of exp(ix) = cos(x) + i sin(x). Plausibility argument via addition formula, proof via Taylor series, or because they satisfy the same differential equation.

  Start of discussion of complex eigenvalues. Even though the general solution involves complex numbers, the solution STAYS REAL if the initial value is real.

  When we have a complex eigenvalue, eigenvalues and eigenvectors come in complex conjugate pairs. The general solution can be rewritten in terms of the real and imaginary parts of a single eigenvector/eigenvalue. This way of writing the general solution allows us to deal only with trig functions.

  Examples and graphs. Complex eigenvalues lead to oscillating solutions. The amplitude of the oscillation can increase/decrease with time, according to whether the real part of the eigenvalue is positive or negative.