JNTU
JNTU B.Tech II MATHEMATICS-III Previous Question Paper (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
For the differential equation x3(x-1)y'' - 2(x-1)y' + 3xy = 0, locate and classify the singular points on the x-axis.
Answer
To find the singular points, write the differential equation in the standard form y'' + P(x)y' + Q(x)y = 0, where P(x) = -2 / (x2(x-1)) and Q(x) = 3 / (x2(x-1)). Singular points occur where the coefficients are not analytic, specifically at x = 0 and x = 1.
For x = 1: Calculate the limits limx→1 (x-1)P(x) = -2 and limx→1 (x-1)2Q(x) = 0. Since both limits are finite, x = 1 is a regular singular point.
For x = 0: Calculate limx→0 xP(x) = limx→0 -2/(x(x-1)) = ∞. Since the limit is infinite, x = 0 is an irregular singular point.
Find the complementary function for x2(d2y/dx2) - 2x(dy/dx) + 3y = 2 sin x log x.
Answer
The given equation is a Cauchy-Euler differential equation of the form x2(d2y/dx2) + Px(dy/dx) + Qy = R(x). To find the complementary function, we substitute x = ez, which implies z = log x. The derivatives are transformed as follows:
where D = d/dz. Substituting these into the homogeneous part of the equation:
The auxiliary equation is m2 - 3m + 3 = 0. Solving for m using the quadratic formula:
Since the roots are complex of the form α ± iβ, where α = 3/2 and β = √3/2, the complementary function in terms of z is:
Substituting back z = log x and ez = x, the complementary function is:
Express the polynomial 1/2 x2 + 10x in terms of Legendre polynomial.
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Prove that integral(J1(x)/x)dx = -J0(x).
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Write down the Cauchy-Reimann equations for f(z) = ez and show that f(z) is analytic for all z.
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Evaluate integral z2 dz from 0 to 3+i along the line y = x2.
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State Cauchy integral Theorem.
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Find the residues at the poles of the function f(z) = 2z(z+3)/((z-3)(z-5)), C: |z| ≤ 3.
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Determine the region of the w-plane into which the first quadrant of the z-plane is mapped under the transformation w = z2.
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Define bilinear transformation. Find the invariant points of bilinear transformation.
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PART-B
Solve in series the equation d2y/dx2 + xy = 0.
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Use Frobenius method to find the solution of 2x2y'' + x(2x+1)y' - y = 0.
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Prove that integral Pn(x)(1-2hx+h2)-1/2 dx = (2hn/(2n+1)) from -1 to 1.
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Prove that d/dx(Jn2(x) + Jn+12(x)) = 2(n/x Jn2(x) - (n+1)/x Jn+12(x)).
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Show that Jn'(x) = 1/2 (Jn-1(x) - Jn+1(x)).
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Prove Rodrigues formula for Legendre polynomials.
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Evaluate the integral (z-1)/(z+1)2 dz over C where C is |z-i|=2.
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If f(z)=u+iv is an analytic function of z, find f(z) if u - v = (x-y)(x2+4xy+y2).
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Find the analytic function whose real part is y/(x2+y2).
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Use Cauchy integral formula to evaluate ez/(z+1)2 dz over C, where C is |z-1|=3.
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Obtain the expression for f(z) = 1/(z2-3z+2) which are valid in the region 1 < |z| < 2. Also name the series thus obtained.
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Evaluate the integral from 0 to infinity of dx/(x4+a4), a > 0 using a suitable contour.
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Find the nature and location of the singularities of the function f(z) = e2z/(z-1)4 by finding its Laurent’s series expansion.
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Use contour integration to evaluate the real integral from 0 to infinity of sin(mx)/(1+x2) dx.
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Find the image of the semi-infinite strip x > 0, 0 < y < 1, under the transformation w = iz+1.
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Find the bilinear transformation that maps the points z1=2, z2=i, z3=-2 into the points w1=1, w2=i, w3=-1, respectively.
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Show that the transformation w = 1/z transforms all circles and straight lines into the circles and straight lines in the w plane.
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Discuss the transformation w = z2. Also find the image of the region |z-2i|=2 under this transformation.
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