Extend the proof of theorem 8.1.4 to an arbitrary positive integer n . Theorem 8.1.4 The set of all solutions to the regular n th -order homogeneous linear differential equation, y ( n ) + a 1 ( x ) y ( n − 1 ) + ⋯ + a n − 1 ( x ) y ′ + a n ( x ) y = 0 on an interval I is a vector space of dimension n .
Extend the proof of theorem 8.1.4 to an arbitrary positive integer n . Theorem 8.1.4 The set of all solutions to the regular n th -order homogeneous linear differential equation, y ( n ) + a 1 ( x ) y ( n − 1 ) + ⋯ + a n − 1 ( x ) y ′ + a n ( x ) y = 0 on an interval I is a vector space of dimension n .
Solution Summary: The author explains the extended proof of theorem 8.1.4 to an arbitrary positive integer n.
Extend the proof of theorem 8.1.4 to an arbitrary positive integer
n
.
Theorem 8.1.4 The set of all solutions to the regular
n
th
-order homogeneous linear differential equation,
y
(
n
)
+
a
1
(
x
)
y
(
n
−
1
)
+
⋯
+
a
n
−
1
(
x
)
y
′
+
a
n
(
x
)
y
=
0
on an interval
I
is a vector space of dimension
n
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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