55. A function f is called homogeneous of degree n if it satisfies the equation f(tx, ty)= tf(x, y) for all t, where n is a positive integer and f has continuous second-order partial derivatives. (a) Verify that f(x, y) = x²y + 2xy2 + 5y³ is homogeneous of degree 3. (b) Show that if f is homogeneous of degree n, then X af af əx ду + y = nf(x, y) [Hint: Use the Chain Rule to differentiate f(tx, ty) with respect to t.]
55. A function f is called homogeneous of degree n if it satisfies the equation f(tx, ty)= tf(x, y) for all t, where n is a positive integer and f has continuous second-order partial derivatives. (a) Verify that f(x, y) = x²y + 2xy2 + 5y³ is homogeneous of degree 3. (b) Show that if f is homogeneous of degree n, then X af af əx ду + y = nf(x, y) [Hint: Use the Chain Rule to differentiate f(tx, ty) with respect to t.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
55 Assume that all the given functions have continuous second-order partial derivatives.
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946
55. A function f is called homogeneous of degree n if it satisfies
the equation
60
f(tx, ty) = tf(x, y)
for all t, where n is a positive integer and f has continuous
second-order partial derivatives.
(a) Verify that f(x, y) = x²y + 2xy² + 5y³ is homogeneous
ed of degree 3.
(b) Show that if f is homogeneous of degree n, then
nf (x, y)
ancisco
कर
CHAPTER 14 Partial Derivatives
[Hint: Use the Chain Rule to differentiate f(tx, ty) with
respect to t.]
56. If f is homogeneous of degree n, show that
a²f
dx²
50
70
+ 2xy
af
dx
Reno
+ y
60
a²f
Əx dy
af
dy
+ y²
on
Las
a²f
ду²
=
14.6 Directional Derivatives and the Gradient Vect
The weather map in Figure 1 shows a
the states of California and Nevada
isothermals, join locations with the s
tion such as Reno is the rate of chang
east from Reno; Ty is the rate of char
Vegas) or in
want to know the rate of change of
n(n − 1)f(x, y)
57. If f is I
58. Suppos
of the t
z = f(-
Fx, Fy,
aitse
59. Equati
defined
is diffe
ond de](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b8c93d7-08d0-4f3d-9fda-3911731791cf%2F60abd683-1ca3-40c0-a8e7-8264e3f45457%2Fpk1p37i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:n
946
55. A function f is called homogeneous of degree n if it satisfies
the equation
60
f(tx, ty) = tf(x, y)
for all t, where n is a positive integer and f has continuous
second-order partial derivatives.
(a) Verify that f(x, y) = x²y + 2xy² + 5y³ is homogeneous
ed of degree 3.
(b) Show that if f is homogeneous of degree n, then
nf (x, y)
ancisco
कर
CHAPTER 14 Partial Derivatives
[Hint: Use the Chain Rule to differentiate f(tx, ty) with
respect to t.]
56. If f is homogeneous of degree n, show that
a²f
dx²
50
70
+ 2xy
af
dx
Reno
+ y
60
a²f
Əx dy
af
dy
+ y²
on
Las
a²f
ду²
=
14.6 Directional Derivatives and the Gradient Vect
The weather map in Figure 1 shows a
the states of California and Nevada
isothermals, join locations with the s
tion such as Reno is the rate of chang
east from Reno; Ty is the rate of char
Vegas) or in
want to know the rate of change of
n(n − 1)f(x, y)
57. If f is I
58. Suppos
of the t
z = f(-
Fx, Fy,
aitse
59. Equati
defined
is diffe
ond de
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