JNTU
JNTU B.Tech II MATHEMATICS-III Previous Question Paper (November/December - 2017 - 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
(25 Marks)
Solve (D4 + 13D2 + 36)y = 0.
Answer
The given differential equation is (D4 + 13D2 + 36)y = 0. The auxiliary equation is m4 + 13m2 + 36 = 0. Letting m2 = x, we have x2 + 13x + 36 = 0, which factors as (x + 9)(x + 4) = 0. Thus, m2 = -9 and m2 = -4, giving roots m = ±3i and m = ±2i. The general solution is y = c1cos(3x) + c2sin(3x) + c3cos(2x) + c4sin(2x).
Find the P.I of (D2 + 4)y = cos 2x.
Answer
To find the Particular Integral (P.I.) of the differential equation (D2 + 4)y = cos 2x, we use the formula for P.I. when the right-hand side is cos(ax):
Here, a = 2. Substituting this into the formula:
Since substituting D2 = -a2 = -4 results in a zero denominator (D2 + 4 = -4 + 4 = 0), we apply the rule for the case of failure:
Prove that Pn+1(x) - Pn-1(x) = (2n+1)Pn(x).
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Prove that ∫0x xJn(x)Jn-1(x) dx = x2/2 [Jn2(x) - Jn-1(x)Jn+1(x)].
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If u = ex(cos y - sin y), then find analytic function f(z).
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Evaluate ∫01+i (x-iy) dz along the path y = x2.
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Expand f(z) = (z+2)/(z-1)(z-3) for |z| > 2.
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Find the residue of f(z) = ez / (z-1)2 at the singular point.
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Find the fixed points of w = (z-1)/(z+1).
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Prove that w = 1/z is circle preserving.
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Part B
(50 Marks)
Solve (D2-1)y = 2sinh x.
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Solve (1+x)2 d2y/dx2 + (1+x) dy/dx + y = 4cos(log(1+x)).
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Obtain the series solution of the equation (x2-4) d2y/dx2 + 2 dy/dx + 2y = 0.
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Prove that ∫-11 Pm(x) Pn(x) dx = 0 if m ≠ n, and 2/(2n+1) if m = n.
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Show that 3P2(x) + 2P3(x) = 5x3.
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Prove that (x2-1)Jn''(x) + xJn'(x) + (x2-n2)Jn(x) = 0.
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Show that J1/2(x) = sqrt(2/πx) sin x.
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Prove that the function f(z) = (x3(1+i)-y3(1-i))/(x2+y2) is continuous and satisfies C-R equations at the origin, yet f'(0) does not exist.
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If f(z) is analytic, prove that (∂2/∂x2 + ∂2/∂y2) |f(z)|2 = 4|f'(z)|2.
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Evaluate ∫c (z+4)/(z2+2z+5) dz where c is the circle (i) |z+1-2i|=2 (ii) |z+1+i|=2.
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State and prove Cauchy’s inequalities.
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State and prove residue theorem.
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Evaluate ∫02π dθ / (a+b cos θ).
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Evaluate ∫0∞ dx / (x4+a4).
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Prove that ∫0∞ (sin mx)/x dx = π/2.
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Plot the image of 1 < |z| < 2 under w = iz+1.
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Find the graph of the region -π < x < π, 1 < y < 2 under w = sin z.
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Find the image of the region in z-plane between y=0 and y=π/2 under w = ez.
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Find the bilinear transformation which maps ∞, i, 0 into -1, -i, 1.
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