DEN - Sample final

You can make it up yourself and solve.

1. Linear equation.
  a) Non-homogeneous linear ODE with RHS that can be handled by guessing.
See practice problems on linear equations.
  b) Additional question that calls on knowledge of lienar differential equations: asymptotic behaviour of solutions at infinity, selection of a certain type of solution by choosing initial conditions, fundamental systems, equation with parameter, transformation of an equation to a system of equations etc.

2. Differential equations.
  a) Solving ODE by separation; see practice problems on separation.
  b) Solving homogeneous systems of linear ODEs or a problem on solving 1st order ODE by variation; see practice problems on systems and variation.

3. Less routine questions.
  a) Vector fields, stability of stationary solutions; see practice problems on analysis.
  b) Existence and uniqueness using the Picard theorem (the version with bounded partial derivative); see practice problems on analysis.
  c) Judging suitability of basic methods for a given ODE; see practice problems on judging methods.
  d) More theoretical or supplementary question from numerical math (solving an ODE using Laplace transform, error in calculations, approximation using Taylor polynomial, algorithm control etc.).

4. Numerical mathematics.
  a): Showing how an algorithm works by doing calculations by hand; see practice problems on numerical math.
  b): Deducing or explaining some important method, discussing its properties (error, order, stopping, pros/cons).

It is recommended that you look at the page called Information on the final.