The final exam consists of two parts, a written test and an optional oral exam.

**Written test:** It will consist of four problems, you will have 90
minutes to complete it. For contents see the file Typical exam.

For the exam, bring some paper to write on and one A4 double-sheet as a container for your written work. You can prepare this cover sheet at home, the front page should include (legible) name and a chart for points:

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To pass the written test one needs to get at least half of the points available (40 out of 80). To pass the exam one has to achieve at least 50 after the points from midterm are added. Then one can improve the grade by going to the oral exam.

**Oral exam:** Proofs of key theorems and facts from the lectures (see
below).

Below you find an overview of things you are expected to master for the exams. For each theme we show topics that a student should be able to do practically and things that he/she should be able to talk about.

**Practical skills:**

Mastering this is the key to passing the written test.
Topics marked by a double dot are the crucial. A student going for
an exam should be able to:

•• find general solution of 1st order ODE by separation

•• find general solution of 1st order linear ODE by variation

•• find general solution of linear ODE of order *n* (homogeneous
using characteristic numbers, RHS using guessing/undetermined coefficients)

•• decide which of the three methods above is suitable for a given
equation

•• derive particular solution from a general one using given initial
conditions

• estimate asymptotic behaviour of solutions at infinity

• select a solution of a desired type by choosing suitable initial
conditions

•• sketch slope field for equations of the type
*y*' = *f* (*x*, *y*)*y*' = *f* ( *y*)

•
for equations of the type
*y*' = *f* (*x*, *y*)

•• find general solution of homogeneous system of linear ODEs of order 1
using eigenvalues (no eigenvalues of higher multiplicity expected, but complex
eigenvalues are possible)

• transform a given differential equation of higher order to a system

• solve a given ODE using Laplace transform (dictionary and grammar
sheet will be supplied)

**Understanding of notions:**

One should understand these notions, be able to express the ideas and
explain. If there is a mark [proof], then the proof of that
statement can be asked for at oral exam.

• what is a linear differential equation

• Theorem on structure of set of all solutions for a homogeneous linear
ODE/system of ODEs [proof]

• what is a fundamental system of a linear differential equation

• Theorem on characteristic numbers yielding solutions of a homogeneous
linear equation [proof]

• Theorem on eigenvalues and eigenvectors yielding solutions
of a homogeneous system of linear ODEs [proof]

• Theorem on structure of set of all solutions for a linear ODE/system
of ODEs (several possible versions) [proof]

**Practical skills:**

This can be in the written test, theoretical topics also in the oral exam.
Topics marked by a double dot are the crucial.

•• integrals: apply (by hand) the rectangle method of a trapezoid method
to a given integral, with given step size or partition size

•• equations: rewrite the given equation into a root finding problem or
a fixed-point problem; show how basic algorithms work
(bisection, Newton, fixed point iteration) for a given problem (do
several iterations by hand)

•• ODE: for a given ODE od order 1 with initial condition, use Euler
method to estimate its solution (do several iterations by hand)

•• systems: given a system of linear algebraic equations, show how
the elimination method (GEM plus backward substitution) works, and show how
Gauss-Seidel iteration works (create iteration formulas, do several
iterations by hand)

•• know how to use practically order of a method (error estimation,
choice of suitable step size)

• when an equation is rewritten as a fixed point problem, guess whether
the resulting iteration has a chance to converge

• when an iterating algorithm produces a number, recognize whether it
is close enough (by the given tolerance) to the one we look for

• deduce an approximating formula of given precision for a given
function and a given center using Taylor expansion

**Understanding of notions, questions of more theoretical nature:**

One should understand these notions, be able to express the ideas and
explain. Could be needed in written test and also at oral exam.

•• integral: know the principle of basic methods of numerical
integration (rectangle, trapezoid), explain with a picture and show
the error of method in it, know how to deduce formulas for rectangle and
trapezoid methods and know their orders

•• roots: know the substance of basic root finding algorithms
(bisection, tangent/Newton, fixed point iteration), be able to write an
algorithm, explain with a picture (bisection, Newton), know advantages and
disadvantages; know the order for bisection and Newton

•• roots: know stopping conditions for iterative algorithms, problem of
achieving desired tolerance, simple test guaranteeing presence of root

•• ODE of order 1: know basic setup for solving them numerically
(partition, going by steps uwing slopes), know the principle of the Euler
method, explain with picture; know its local and global error and its order

• ODE of order 1 (for advanced students): how Runge-Kutta methods of
higher order work (basic principle)

• systems: GEM: principle, computational complexity, pivoting and why
is it done, how we find solution using back substitition (complexity);
GSM: fixed point for systems, computational complexity, convergence

• the idea behind methods for approximating derivative, explain
the three basic methods with a picture, know their order of error;
deduce the error of the forward difference using Taylor expansion

• in general: for every numerical task (derivation,
integration, finding root by iteration, solving initial value problem for
ODE) one should know how to judge quality of methods: principle
of the notion of order (for integrals
*C**h*^{q}*C*/*n*^{q},

•• absolute and relative error, error propagation, (relatively)
safe and unsafe operations; know how to
derive estimate of absolute and relative error for simple operations
(addition, product, for advanced students also others)

• know what errors appear in real-life calculations, where they come
from, how they behave, problems when calculating in floating point
(round-off error, sometimes shift in exponents, zero is not precise),
what is numerical stability

Concerning definitions and theorems, it is not necessary to know the statements by heart precisely as given, one can use also other formulations. The crucial thing here is that whatever you say must capture the correct meaning, so the logical contents must be preserved.