JNTU
JNTU B.Tech II MATHEMATICS-III Previous Question Paper (December - 2018 - R13)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b, c as sub questions.
Part A
Determine the nature of the point x = 0 for the equation x(x-1)3 y'' + 2(x-1)3 y' + 3y = 0.
Answer
Rewrite the equation as y'' + P(x)y' + Q(x)y = 0, where P(x) = 2(x-1)3 / [x(x-1)3] = 2/x and Q(x) = 3 / [x(x-1)3]. At x = 0, P(x) is undefined, so x = 0 is a singular point. Since lim(x→0) xP(x) = 2 and lim(x→0) x2Q(x) = lim(x→0) 3x/(x-1)3 = 0, both limits are finite. Thus, x = 0 is a regular singular point.
Find the C.F of (x2D2 + 3xD + 1)y = 1/(1-x)2.
Answer
This is a Cauchy-Euler equation. Let x = et, so t = ln(x). The equation becomes (D'(D'-1) + 3D' + 1)y = f(et), where D' = d/dt. This simplifies to (D'2 + 2D' + 1)y = 0. The characteristic equation is m2 + 2m + 1 = 0, giving m = -1, -1. The complementary function is C.F. = (c1 + c2t)e-t = (c1 + c2ln(x))/x.
Express 2 - 3x + 4x2 in terms of Legendre polynomial.
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Define i) Entire function ii) Harmonic function.
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Show that the function f(z) = z/|z| is not continuous at z = 0.
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Find the radius of convergence of Σ (z-i)n/(n+1)2.
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Find the residue of the function sin(z)/z at z=0.
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Find the invariant points of the transformation w = z2.
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Determine the bilinear transformation whose fixed points are 1, -1.
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Part B
Obtain power series solution of y'' + (x-1)y' + y = 0 in power of (x-2).
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Solve (x+1)(x+2)y'' + (x+3)y' + y = 0.
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Show that when n is a positive integer, Jn(x) is the co-efficient of zn in the expansion of exp{x(z-1/z)/2}.
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Express x2 - 3x + 7 in terms of Legendre Polynomial.
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Evaluate ∫(x-iy)dz along y=x from 0 to 1+i.
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Evaluate ∮ dz/(z2+9) where |z-i|=4 by Cauchy’s integral formula.
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Find the analytic function whose real portion is e-x(x cos y + y sin y).
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Evaluate ∮ (z3+z2+4)/(z2+4z+8) dz where C is the circle |z-2|=4 clockwise.
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Evaluate ∮ z2/(z2+2z+5) dz for: a) |z|=1, b) |z+1-i|=2, c) |z+1+i|=2.
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Expand f(z) = (z-1)/(z+1) in Taylor’s series about the point a) z = 0 and b) z = 1.
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Find the image of x=2 under the transformation w = z2.
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Find the bilinear transformation that maps z = 1, i, -1 onto w = i, 1, 0.
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Find the image of the region 0 < x < 2π, 1 < y < 2 in the z plane under the mapping w = sin z.
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Determine the Mobius transformation having 1 and i as fixed points and maps 0 to -1.
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