determine
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Show me the steps of determine red and it complete
![8.3.2 Example B
Consider the following Cauchy-Euler differential equation
dy
+ y = 0.
(8.77)
dx?
Its characteristic equation is
r(r – 1) +1= r² – r +1 = 0,
(8.78)
-
with solutions
V3
i,
2
1
ri = r2
2
i = V-1.
(8.79)
To obtain the general solution, the quantity x"1 must be calculated. This is
done as follows
(x ) exp
+
i In x
2
(8.80)
= x
Therefore, y(x) is given by the expression
[(4)
[(4)-}
72 A cos
In x + B sin
In x
(8.81)
= X
Note that for x > 0, the solution oscillates with increasing amplitude.
The corresponding discrete version of equation (8.77) is
k(k + 1)A²yk + Yk = 0,
(8.82)
and its characteristic equation is that given in equation (8.78). Therefore, the
general solution is
Yk = Ck+r) + c•T(k+r*)
r(k +r*)
+ C*
I(k)
(8.83)
I(k)
where r = r1 and C is an arbitrary complex number. Observe that the manner
in which the right side of equation (8.83) is written insures real values for Yk-
This depends also on the fact that the gamma function I(z) is real-valued,
i.e., for z = x + iy, I'(z*) = [T'(2)]*. The integral representation of the gamma
function allows this to be easily demonstrated, i.e.,
T(z)
e-ttz-1dt.
(8.84)
Finally, while it is not to be expected that y(x), equation (8.81), and yk,
equation (8.83) have "exactly" the same mathematical structure for all x and
k, where the correlation between these variables is
x → Xk = (Ax)k, Ax = 1,
(8.85)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06b6425a-ab8a-4368-90d9-64663800e88c%2F127b580d-e00c-46fd-82c4-95000716cd88%2Fne9g9i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:8.3.2 Example B
Consider the following Cauchy-Euler differential equation
dy
+ y = 0.
(8.77)
dx?
Its characteristic equation is
r(r – 1) +1= r² – r +1 = 0,
(8.78)
-
with solutions
V3
i,
2
1
ri = r2
2
i = V-1.
(8.79)
To obtain the general solution, the quantity x"1 must be calculated. This is
done as follows
(x ) exp
+
i In x
2
(8.80)
= x
Therefore, y(x) is given by the expression
[(4)
[(4)-}
72 A cos
In x + B sin
In x
(8.81)
= X
Note that for x > 0, the solution oscillates with increasing amplitude.
The corresponding discrete version of equation (8.77) is
k(k + 1)A²yk + Yk = 0,
(8.82)
and its characteristic equation is that given in equation (8.78). Therefore, the
general solution is
Yk = Ck+r) + c•T(k+r*)
r(k +r*)
+ C*
I(k)
(8.83)
I(k)
where r = r1 and C is an arbitrary complex number. Observe that the manner
in which the right side of equation (8.83) is written insures real values for Yk-
This depends also on the fact that the gamma function I(z) is real-valued,
i.e., for z = x + iy, I'(z*) = [T'(2)]*. The integral representation of the gamma
function allows this to be easily demonstrated, i.e.,
T(z)
e-ttz-1dt.
(8.84)
Finally, while it is not to be expected that y(x), equation (8.81), and yk,
equation (8.83) have "exactly" the same mathematical structure for all x and
k, where the correlation between these variables is
x → Xk = (Ax)k, Ax = 1,
(8.85)
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