Exam (elaborations)
MAT3700 EXAM PACK 2023
- Institution
- University Of South Africa
MAT3700 EXAM PACK 2023
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MAT3700EXAM
PACK 2023
LATEST QUESTIONS AND
ANSWERS
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MAT3700 Outcome:
Outcome 1:
Solving first order differential equations
Assessment criteria:
1. Use direct integration, separation of variables, substitution and the integrating factor method to
solve exact, linear, Bernoulli and homogeneous first-order differential equations.
2. Apply knowledge to solve practical problems involving growth and decay, cooling, mixtures and
falling bodies.
Outcome 2:
Solving second order differential equations of the form
is equal to zero, a constant, or a function of x only,
and P0, P1 and P2 are constants.
Assessment criteria
1. Use the method of undetermined coefficients to find the general solution of second order
differential equations.
2. Be able to find the particular solution from the general solution when given the necessary
conditions.
Outcome 3:
Solving second order differential equations of the form
is equal to zero, a constant, or a function of x only,
and P0, P1 and P2 are constants and n is a natural number. assessment criteria.
1. Use D-operator methods to find the general or particular solution.
2. Use Laplace transforms and inverse Laplace transforms to find the particular solution.
Outcome 4:
Determine eigenvalues and eigenvectors of a matrix
Assessment criteria
1. Be able to calculate the eigenvalues of a 2x2 or 3x3 matrix.
2. Given an eigenvalue of a 2x2 or 3x3 matrix be able to find an eigenvector corresponding to the
eigenvalue.
Outcome 5:
Representing a function as a Fourier series.
Assessment criteria
1. Be able to sketch a function over a given range and expand the sketch to represent a periodic
function.
2. Obtain the Fourier series expansion of the periodic function.
Two topics are not covered in your study guides: Linear
Algebra and Fourier Series. To master this content you need to buy the prescribed book named
in paragraph 4 of this letter. The prescribed book will also provide more examples and
exercises on the other topics in this module.
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The word Calculus comes from Latin meaning "small stone", Because it is like understanding
something by looking at small pieces.
Differential Calculus cuts something into small pieces to find how it changes.
Integral Calculus joins (integrates) the small pieces together to find how much there is.
Solving differential equations:
Separable y’ = f(x).g(y)
Homogeneous y’ = (y/x)
Integrating Factor
Exact M(x,y)dx+N(x,y)dy = 0 M(x) = N(y)
Linear y’ + p(x)y = q(x)
Direct y’ = f(x) and y = ∫f(c)dx + c
Substitution dy dy du
dx = du dx
Classify Before Trying To Solve: Ordinary or Partial
"Ordinary Differential Equations" (ODEs) have a single independent variable (like y)
"Partial Differential Equations" (PDEs) have two or more independent variables.
Ordinary Differential Equations
Order and Degree The Order is the highest derivative (is it a first
derivative? a second derivative? etc)
The degree is the exponent of the highest
derivative.
Linear
It is Linear when the variable (and its derivatives) has no exponent or other function put on it.
So no y^2, y^3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is). dy/dx + P(x)y = Q(x)
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