XXXX 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.
XXXX 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.
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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Question
[Partial
The second picture is for context
![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%2Fd7a4e2dd-fef9-462f-a1aa-125d5a5927c0%2F0e800dd9-27fa-4dfb-805a-f2629624efb4%2Focfz87_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.
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)
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