(a) If X ( t ) = ( x 11 ( t ) x 12 ( t ) x 21 ( t ) x 22 ( t ) ) and u ( t ) = ( u 1 ( t ) u 2 ( t ) ) , show that ( X u ) ′ = X ′ u + X u ′ . (b) Assuming that X ( t ) is a fundamental matrix for x ' = P ( t ) x and that u ( t ) = ∫ X − 1 ( t ) g ( t ) d t , use the result of part (a) to verify that x p ( t ) given by Eq. ( 15 ) satisfies Eq. ( 1 ) , x ' = P ( t ) x + g ( t ) .
(a) If X ( t ) = ( x 11 ( t ) x 12 ( t ) x 21 ( t ) x 22 ( t ) ) and u ( t ) = ( u 1 ( t ) u 2 ( t ) ) , show that ( X u ) ′ = X ′ u + X u ′ . (b) Assuming that X ( t ) is a fundamental matrix for x ' = P ( t ) x and that u ( t ) = ∫ X − 1 ( t ) g ( t ) d t , use the result of part (a) to verify that x p ( t ) given by Eq. ( 15 ) satisfies Eq. ( 1 ) , x ' = P ( t ) x + g ( t ) .
X
(
t
)
=
(
x
11
(
t
)
x
12
(
t
)
x
21
(
t
)
x
22
(
t
)
)
and
u
(
t
)
=
(
u
1
(
t
)
u
2
(
t
)
)
,
show that
(
X
u
)
′
=
X
′
u
+
X
u
′
.
(b) Assuming that
X
(
t
)
is a fundamental matrix for
x
'
=
P
(
t
)
x
and that
u
(
t
)
=
∫
X
−
1
(
t
)
g
(
t
)
d
t
, use the result of part (a) to verify that
x
p
(
t
)
given by Eq.
(
15
)
satisfies Eq.
(
1
)
,
x
'
=
P
(
t
)
x
+
g
(
t
)
.
The total-revenue and total-cost funtions for producing x clocks are R(x)=500x-0.01x2 and C(x)=160x+100,000, where x is greater than equal zero and less than equal 25,000. What is the maximum annual profit?
A particular solution of x¨ − 3x˙ + 2x = t2 + 1 is given by
Find the critical points of the function f(x)=x2−4x+3x−2.
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Differential Equations: An Introduction to Modern Methods and Applications
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RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY