11 (a) Find the analytic function f (z) = u + iv where u(x, y) =e* (xcosy - ysiny). 6z +z (b) Evaluate dz where C is the circle z-1= 1. z -1 C.
11 (a) Find the analytic function f (z) = u + iv where u(x, y) =e* (xcosy - ysiny). 6z +z (b) Evaluate dz where C is the circle z-1= 1. z -1 C.
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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Please can u send the solution of these problems please it will be great help plzz I request u to send all the question solutions plzz .you can send the solution one by one plzzzz
![11. a) Find the analytic function f(z)=u+iv, if u-v = (x – y\x² + 4.xy+ y² ).
15. A homogeneous rod of conducting material of length 100 cm has its ends kept at
0SxS 50
(11
b) Evaluate
sin z
+ coS TZ
-dz where C is the circle z= 3.
x,
(z – 1)(z – 2)
zero temperature and the temperature initially is u (x, 0)=
100-x 50sIS100
Find the temperature u(x,1) at any time.
12. a) Find Laurent's expansion of
7z- 2
z(z+1)(z – 2)
x²
b) Evaluate +1)x* + 4)
f(2) =
in the region 1<|z+ 1|< 3.
16. a) Find the bilinear transformation which maps the points z = 1, i,-1 onto the points
'E > + z|> 1
w = i,0,-i.
8.
dx.
b) Evaluate
dz where C is the circle |z – 2| = ÷.
(z- 1Xz- 2)
13. If f(x)=cos x), expand f(x)as a Fourier series in the interval (-,7).
-2-
2e2* +3x y.
%3D
17. a) Solve
14. a) Solve (x? - y² - z² )p+ 2xyq = 2xz.
b) Solve 2z + p² + qy+ 2y = 0 by using Charpit's method.
%3D
b) Find the complete solution of z (p² +q° )= x² + y²,
3
%3D
13 a) Find the Fourier series expression of f(x) = x + x, -T S X ST.
1
Deduce that 1+
1
1
22 32 42
b) Obtain the half range Fourier sine series for f(x) = 2x – 1,0 < x < 1.
14 a) Obtain the general solution of 2xzp + 2yzq + x² + y - z? = 0.
b) Solve p + q = pq by Charpit's method.
11 a) Show that the function u(x, y) = 4xy - 3x+2 is harmonic and find its conjugate
%3D
harmonic function v(x, y).
15 Solve the Laplace equation-
,2
dy
= 0 for a rectangular plate subject to the following
z-1
b) Apply Cauchy's integral formula to evaluate
dz, where C is |z-i| = 2.
(z+1)2 (z- 2)
conditions.
U(x, 0) = u(x, 1) = 0, u(0, y) = y and
ди
- (1, y) = -5
z2 -1
12 a) Expand f(z) =
in the regions (i) Iz| < 2 and (ii) 2 < |z| < 3.
(z+2)(z+3)
16 a) Find the analytic function f(z) such that Re[f (z)] = 3x² – 4y – 3y² and f(1+i) = 0.
%3D
%3D
2x
b) Evaluate|
1+2 cos0
de.
b) Find the bilinear transformation which maps the points z = 1, i, -1 into w = i, 0, -i and
hence find the image of |z| <.1.
5+4cos0
17 a) Solve [2D² – 5DD' + 3(D')² +D- D'] z = e*y.
TC
Cos
cos
cos
8
b) Show that |x| = 1--
-2 < x < 2.
%3D
32
52
TC
1604](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc17eed4c-0d50-4418-9922-c1f097e88630%2F7d376c61-642b-4e34-9f24-26051e82fdca%2F5smfgpa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:11. a) Find the analytic function f(z)=u+iv, if u-v = (x – y\x² + 4.xy+ y² ).
15. A homogeneous rod of conducting material of length 100 cm has its ends kept at
0SxS 50
(11
b) Evaluate
sin z
+ coS TZ
-dz where C is the circle z= 3.
x,
(z – 1)(z – 2)
zero temperature and the temperature initially is u (x, 0)=
100-x 50sIS100
Find the temperature u(x,1) at any time.
12. a) Find Laurent's expansion of
7z- 2
z(z+1)(z – 2)
x²
b) Evaluate +1)x* + 4)
f(2) =
in the region 1<|z+ 1|< 3.
16. a) Find the bilinear transformation which maps the points z = 1, i,-1 onto the points
'E > + z|> 1
w = i,0,-i.
8.
dx.
b) Evaluate
dz where C is the circle |z – 2| = ÷.
(z- 1Xz- 2)
13. If f(x)=cos x), expand f(x)as a Fourier series in the interval (-,7).
-2-
2e2* +3x y.
%3D
17. a) Solve
14. a) Solve (x? - y² - z² )p+ 2xyq = 2xz.
b) Solve 2z + p² + qy+ 2y = 0 by using Charpit's method.
%3D
b) Find the complete solution of z (p² +q° )= x² + y²,
3
%3D
13 a) Find the Fourier series expression of f(x) = x + x, -T S X ST.
1
Deduce that 1+
1
1
22 32 42
b) Obtain the half range Fourier sine series for f(x) = 2x – 1,0 < x < 1.
14 a) Obtain the general solution of 2xzp + 2yzq + x² + y - z? = 0.
b) Solve p + q = pq by Charpit's method.
11 a) Show that the function u(x, y) = 4xy - 3x+2 is harmonic and find its conjugate
%3D
harmonic function v(x, y).
15 Solve the Laplace equation-
,2
dy
= 0 for a rectangular plate subject to the following
z-1
b) Apply Cauchy's integral formula to evaluate
dz, where C is |z-i| = 2.
(z+1)2 (z- 2)
conditions.
U(x, 0) = u(x, 1) = 0, u(0, y) = y and
ди
- (1, y) = -5
z2 -1
12 a) Expand f(z) =
in the regions (i) Iz| < 2 and (ii) 2 < |z| < 3.
(z+2)(z+3)
16 a) Find the analytic function f(z) such that Re[f (z)] = 3x² – 4y – 3y² and f(1+i) = 0.
%3D
%3D
2x
b) Evaluate|
1+2 cos0
de.
b) Find the bilinear transformation which maps the points z = 1, i, -1 into w = i, 0, -i and
hence find the image of |z| <.1.
5+4cos0
17 a) Solve [2D² – 5DD' + 3(D')² +D- D'] z = e*y.
TC
Cos
cos
cos
8
b) Show that |x| = 1--
-2 < x < 2.
%3D
32
52
TC
1604
Transcribed Image Text:11 (a) Find the analytic function f (z) = u + iv where u(x, y) =e* (xcosy - ysiny). 2
6z2 +z
(b) Evaluate 02
dz where C is the circle z-1= 1.
z2 -1
|
12 (a) Expand f(z)=
1
in a Laurent series valid for (i)1</z| <3 (ii) z>3
(z +1)(z+ 3)
dx
(b) Evaluate
„(l+x²)?
14 (a) Solve z(p-q)=z²+(x+y)²
(b) Solve 2(z + xp + yp) =yp2 by using Charpits method.
du
15 Find the solution of the heat equation
a²u
0<x<l,t>0 subject to the condition
at
u(0, t) = 0 = u(?, t) and u(x, 0) = &x - x².
Əx²
16 (a) Find the bilinear transformation which maps the points 1, i, -1 onto the points
604
0,1,00.
(b) Evaluate (cos(2°) where C is the circle z=2
cos(z')
(z +1)?
17 (a) Solve (D² + DD' – 6D² ) = y sin x
-
(b) Find the complete and singular integrals of z = px + qy - 2 pq
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