Download Systems of equations, vectors and matrices

Systems of equations, vectors and matrices
We are going to investigate systems of differential equations. A system of two equations, in two unknowns
x(t) and y(t), has the following form:
(
dx/dt = f (t, x, y)
dy/dt = g(t, x, y)
Systems (possibly with more variables and equations) are used to describe electrical circuits, mechanical
systems, interacting populations, and much much more.
One place where we will use systems is when we look at higher order equations. For example, if u00 =
f (t, u, u0 ) is any second-order differential equation then we can study this as a system by considering u and
its derivative as the components. In more detail, if we set v = u0 then v 0 = u00 = f (t, u, v), and so we obtain
the system
(
du/dt = v
dv/dt = f (t, u, v)
We look at the two most important types of systems of differential equations.
1. Autonomous systems:
(
dx/dt = f (x, y)
dy/dt = g(x, y)
We can convert such a system to a single first-order differential equation, by using the Chain Rule to eliminate
t altogether:
dy
dy/dt
g(x, y)
=
=
.
dx
dx/dt
f (x, y)
Note that the independent variable in this equation is x. One benefit of this reduction is that the slope field
for the first-order equation can be used to derive the direction field for the autonomous system. Specifically,
in the xy-plane we can record at each point a vector (arrow) whose x- and y-components are dx/dt and
dy/dt, respectively. For example, one such velocity vector might look like this:
velocity vector =
dx/dt
dy/dt
=
−2
3
dy/dt = 3
dx/dt = −2
The slope of this arrow is precisely dy/dx. Notice however that there is more information in the vector than
just the slope: the head of the arrow tells us whether dx/dt is positive, negative, or zero, and simultaneously
also for dy/dt. Note also that in this picture, t does not appear explicitly. The integral curves in the direction
field are sometimes called trajectories. They are parametric curves. When you look at them you should see
a point in motion, not a static curve.
2. Linear systems:
(
dx/dt = ax + by + f
dy/dt = cx + dy + g
where a, b, c, d, f , and g are all functions of t. These functions might be constant or nonconstant. In many,
but not all, cases the coefficients a, b, c, and d are constant, whereas the components of the driving function
f and g are not. When f and g are identically 0 we say that the equation is homogeneous.
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Especially with linear systems it is convenient to use a different notation, namely that of vectors and
matrices. As we have seen a vector is just a quantity with both a magnitude and direction, and can be
expressed as a list of numerical components. In two dimensions this vector has two components, and we
visualize the vector as in the example above. The most important algebraic operation with vectors is the
linear combination. For example, if
5
2
~v1 =
, ~v2 =
,
−1
4
then the linear combination 3~v1 − 2~v2 is computed as follows:
5
2
15
4
11
3
−2
=
−
=
.
−1
4
−3
8
−11
We can also write this linear combination as an example of matrix multiplication. A matrix is simply a
rectangular array of numbers, which we often regard as a list of column vectors. Thus, if we put the above
vectors ~v1 and ~v2 into a matrix, and put the scalars 3 and −2 into another vector, then we can write the
above linear combination in the form
5 2
3
11
=
.
−1 4 −2
−11
Again, this is just another notational shorthand:
computed above.
We can extend the matrix multiplication a bit.
5
−1
the real algebraic content is the linear combination as
For example, to compute the matrix product
2 −2 1
4
3 1
we use each of the columns of the second matrix to compute linear combinations of the columns of the first
matrix:
5 2 −2 1
−10 + 6 5 + 2
−4 7
=
=
.
−1 4
3 1
2 + 12 −1 + 4
14 3
We can use matrix notation to express a system of linear differential equations. If we let
a b
x
f
,
A=
, w
~=
, and F~ =
c d
y
g
then the system
(
dx/dt = ax + by + f
dy/dt = cx + dy + g
can be written more succinctly as w
~ 0 = Aw
~ + F~ .
Further reading
The text introduces systems of differential equations in section 7.1, and vector and matrix algebra in section
7.2. The text moves on to eigenvectors and eigenvalues on section 7.3.
Reading quiz
1. What is the difference between a slope field and a direction field?
2. What is meant be a linear combination of vectors? Give examples.
3. How is matrix multiplication defined in terms of linear combinations?
4. What is an autonomous system of differential equations?
5. How do we convert a second-order differential equation to a system of differential equations? Give an
example.
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Exercises
Convert the following differential equations into systems. Write the systems in matrix form. If the system
is autonomous then (a) convert it into a first-order differential equation; (b) draw its direction field; and (c)
solve the first-order equation (you may need to review the method from problem 30, pages 46–47).
1. u00 + 5u0 − 4u = 0.
2. u00 = 2 sin(u).
3. u00 + uu0 = 0.
4. u00 + 4u = cos(t).
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