2. Which of the following operators are linear? (a) Lu=ux + xUy (b) Luux + uu y (c) Lu= ux+u ²₂ (d) Lu=ux + Uy + 1 Ux (e) £u=V1+x2 (cos y)u. +uỵry — [arctan(.x/y)]u
2. Which of the following operators are linear? (a) Lu=ux + xUy (b) Luux + uu y (c) Lu= ux+u ²₂ (d) Lu=ux + Uy + 1 Ux (e) £u=V1+x2 (cos y)u. +uỵry — [arctan(.x/y)]u
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
[Partial
The second picture is for context
Transcribed Image Text:Linearity means the following. Write the equation in the form Lu = 0,
where L is an operator. That is, if v is any function, Lu is a new function. For
instance, L = a/ax is the operator that takes v into its partial derivative Ux.
In Example 2, the operator L is L=ə/Əx yə/Əy. (Lu
a/ax +ya/ay. (Lu = ux + yuy.) The
definition we want for linearity is
+
L(u + v) = Lu + Lv
L(cu) = cLu
(3)
for any functions u, v and any constant c. Whenever (3) holds (for all choices
of u, v, and c), L is called linear operator. The equation
Lu = 0
(4)
is called linear if L is a linear operator. Equation (4) is called a homogeneous
linear equation. The equation
Lu = 8,
(5)
where g 0 is a given function of the independent variables, is called an
inhomogeneous linear equation. For instance, the equation
(cos xy²)ux - y²uy = tan(x² + y²)
is an inhomogeneous linear equation.
(6)
![2. Which of the following operators are linear?
(a) Lu=ux + xUy
(b) Lu=ux + uu y
(c) Lu= ux+u²
(d)
(e) Lu= √
Lu=ux + Uy + 1
V1+x2 (cos y)u. +uyxy – [arctan(x/y)]u
-
3. For each of the following equations, state the order and whether it
is nonlinear, linear inhomogeneous, or linear homogeneous; provide
reasons.
(a) ut -Uxx + 1 = 0
Ut
(b) ut -Uxx + xu = 0
(c) UtUxxt + uux = 0
(d) utt - Uxx + x² = 0
(e) iu uxx+u/x = 0
= 0
(f) ux(1+u²)¯¹⁄² +uy(1 +u²)¯¹⁄² ;
(g) ux+e³uy = 0
(h) U₁+Uxxxx + √√1+u = 0
4. Show that the difference of two solutions of an inhomogeneous linear
equation Lu = g with the same g is a solution of the homogeneous
equation Lu= 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45b359f0-fb4a-47fb-b3b7-86deec526e99%2Ffbfed480-1dac-47a2-9b40-13eb5a359d8e%2F0h8ggve_processed.png&w=3840&q=75)
Transcribed Image Text:2. Which of the following operators are linear?
(a) Lu=ux + xUy
(b) Lu=ux + uu y
(c) Lu= ux+u²
(d)
(e) Lu= √
Lu=ux + Uy + 1
V1+x2 (cos y)u. +uyxy – [arctan(x/y)]u
-
3. For each of the following equations, state the order and whether it
is nonlinear, linear inhomogeneous, or linear homogeneous; provide
reasons.
(a) ut -Uxx + 1 = 0
Ut
(b) ut -Uxx + xu = 0
(c) UtUxxt + uux = 0
(d) utt - Uxx + x² = 0
(e) iu uxx+u/x = 0
= 0
(f) ux(1+u²)¯¹⁄² +uy(1 +u²)¯¹⁄² ;
(g) ux+e³uy = 0
(h) U₁+Uxxxx + √√1+u = 0
4. Show that the difference of two solutions of an inhomogeneous linear
equation Lu = g with the same g is a solution of the homogeneous
equation Lu= 0.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
