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Unformatted text preview: x form: dx
2
= t x − y + cos(2t)
dt
dy
3
= x + 4 sin(t)y + t
dt
2
dx
t
−1
x
cos(2t)
=
+
y
1 4 sin(t)
y
t3
dt Introduction to systems of equations
• Any linear system can be written in matrix form: dx
2
= t x − y + cos(2t)
dt
dy
3
= x + 4 sin(t)y + t
dt
2
dx
t
−1
x
cos(2t)
=
+
y
1 4 sin(t)
y
t3
dt
• We’ll focus on the case in which the matrix has constant entries. And
homogeneous, to start. For example,
dx
11
x
=
41
y
dt y Introduction to systems of equations
• Geometric interpretation  direction ﬁelds.
dx
11
x
=
41
y
dt y Introduction to systems of equations
• Geometric interpretation  direction ﬁelds.
dx
11
x
=
41
y
dt y or x = Ax
Introduction to systems of equations
• Geometric interpretation  direction ﬁelds.
dx
11
x
=
41
y
dt y or x = Ax
• Think of the unknown functions as coordinates (x(t), y (t)) of an
object in the plane. Introduction to systems of equations
• Geometric interpretation  direction ﬁelds.
dx
11
x
=
41
y
dt y or x = Ax
• Think of the unknown functions as coordinates (x(t), y (t)) of an
object in the plane.
• Ax gives the velocity vector of the object located at x. Introduction to systems of equations
• Geometric interpretation  direction ﬁelds.
dx
11
x
=
41
y
dt y or x = Ax
• Think of the unknown functions as coordinates (x(t), y (t)) of an
object in the plane.
• Ax gives the velocity vector of the object located at x. Introduction to systems of equations
• Geometric interpretation  direction ﬁelds.
dx
11
x
=
41
y
dt y or x = Ax
• Think of the unknown functions as coordinates (x(t), y (t)) of an
object in the plane.
• Ax gives the velocity vector of the object located at
2
x=
1 x. Introduction to systems of equations
• Geometric interpretation  direction ﬁelds.
dx
11
x
=
41
y
dt y or x =...
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 Spring '13
 EricCytrynbaum
 Differential Equations, Systems Of Equations, Eigenvectors, Equations, Vectors

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