You will see many techniques for solving differential
equations in the course. Each technique really
involves little more than using pre -calculus algebra, calculus, and some linear
algebra in the right
order. The “hard” part is learning and remembering the sequence of steps in each
method , and
the “easy” part should be the calculus and algebra involved in each step. Make
sure you always
check your algebra, it is very easy to turn a straightforward question into a
very difficult question
by making a careless algebra error early on. Here are some topics from
pre-requisite courses which
you should be comfortable with.
(1) Functions and independent variables: The use of different letters can cause
confusion,
particularly x. You should always establish which letter(s) are functions, and
which is the
variable before you attempt to solve any problem. Probably the easiest “rule of
thumb” is
that you should only have derivatives of functions, and there is probably only
one variable
which you may have to name yourself. For example in the equation x ' = tx, t must
be
the variable and x is a function of t. In the equation y' = y^{2} you can choose
any letter for
the variable, other than y which denotes the function. In the system y' = x, x'
= y both
x and y are functions, and usually we would use t for the variable.
(2) Logarithms : If then
. Remember that ln(a+b) ≠ ln(a)+ln(b).
(3) Exponentials : Make sure you understand the rules for exponents, this is
particularly
important when finding integrating factors. For example
and .
(4) Factoring Polynomials : You should know how to factor polynomials and find
their
roots, this is needed for characteristic equations and also for finding
eigenvalues. For
quadratic polynomials there is the quadratic formula, but for higher order
polynomials
the formulas are either very, very complicated or non-existent. Instead you
should look
for “obvious” roots (0, 1, 2 − 1,−2 etc.) and divide out the corresponding
factors using
long division, and then repeat the process if necessary until you find all the
roots. For
example, −1 is a root of r ^{3} + 1 because (−1)^{3} + 1 = 0 and so r + 1 is a factor
of r^{3} + 1.
Long division gives r^{3} + 1 = (r + 1)(r^{2} − r + 1) and then you find the roots of
r^{2} − r + 1
using the quadratic formula. A polynomial will have as many roots as its degree,
though
some may be repeated roots.
(5) Integrals: Understand how to do integrals by substitution, by parts, and
using partial
fractions. You should know how to find anti-derivatives of functions such as
tan(x), xe^{x}
and 1/(x^{2} − x) without using a calculator.
Do not forget the constant(s) of integration. Sometimes we assume for simplicity
that
constants are 0 or 1 but this is ONLY when we are looking for ONE function (for
example,
finding an integrating factor or solving the Variation of Parameters equations).
A general
solution always contains constant(s).
(6) Matrices: You should know how (and when) to multiply matrices and vectors
and the
dimensions of the solution. You should be able to calculate the determinant of
any
square matrix using the row operations /triangular form method and also by the
cofactor/
expansion formulas.
A large part of the second half of the course involves finding eigenvalues
(solve the polynomial
equation det(A−λI) = 0 to find λ ) and eigenvectors corresponding the eigenvalue
λ
(solve the vector equation (A −λI)v = 0 to find v) so you need to understand
these
concepts from linear algebra .