0800MAT201122003

Reg No.:_. Name:

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
Third Semester B.Tech Degree Examination December 2020 (2019 Scheme)

Course Code: MAT201
Course Name: Partial Differential equations and Complex analysis

Max. Marks: 100 Duration: 3 Hours
PARTA
Answer all questions. Each question carries 3 marks Marks
1 Form the PDE for the equation 2 = f(x? ‏(2بر-‎ where f is an arbitrary (3)
function.
2 Solve ‏شد‎ + 2 = 0 given that when x = 0,2 = 0? and 22-1 (3)
0x2 Ox

3 Write the conditions in which a tightly stretched string of length 1 with fixed (3)
end is initially in equilibrium position and is set vibrating by giving each

⋅ ⋅ 8 71%
point a velocity 1057713 ४

4 Write down the possible solutions of one dimensional heat equation. (3)

5 Find the real part and imaginary part of the function f(z) = 522 - 122 + (3)
3 + 2i and find their values at z = 4 — 34

6 Find the fixed points of the mapping w = (a + ib)z? (3)

7 Evaluate ಕ್ಷೆ = dz where © is |2| = 3 (3)

2-2

8 Find the Maclaurin series expansion of ந்‌ ९“ धा (3)

9 Find the Laurent series of z~>sinz with centre 0 (3)

10 Find the residue at poles for the function f(z) = = (3)
PART B

Answer any one full question from each module. Each question carries 14 marks
Module 1
11(a) Solve (x? — 92 —z?)p + 2xyq = 2xz (7)

(b) Solve (p? + 12)» = ரச by Charpit’s method 0

12(a) Find the differential equation of all planes which are at a constant distance ‘a’ (7)

from the origin
(0) 80146 32 +2 > = 0 where u(x,0) = 467% by the method of separation of (7)

variables.

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