JNTU
JNTU B.Tech II MATHEMATICS-III Previous Question Paper (November/December - 2018 - R15)
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.
Part A
Find the complementary function of x^2 d^2y/dx^2 - 3x dy/dx + 4y = x^2 log x.
Answer
The given equation is a Cauchy-Euler equation of the form
Find the P.I of (x^2 d^2y/dx^2 + 3x dy/dx + y) = (1-x)^-2.
Answer
The given equation is a Cauchy-Euler equation of the form (x2D2 + 3xD + 1)y = (1-x)-2. To solve this, we use the substitution x = ez, which implies z = ln(x). The operator xD is replaced by D' and x2D2 is replaced by D'(D'-1), where D' = d/dz.
Substituting these into the differential equation:
This simplifies to (D'+1)2y = (1-ez)-2. The Particular Integral (P.I.) is given by:
Using the method of variation of parameters or the formula for repeated roots, the P.I. is:
Evaluating the inner integral, ∫ ez(1-ez)-2 dz = (1-ez)-1. Then, the outer integral becomes ∫ ez(1-ez)-1 dz = -ln(1-ez). Substituting back z = ln(x):
Prove that P_n(-x) = (-1)^n P_n(x).
Answer
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Evaluate ∫ x^-1 J_0(x) dx.
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Prove that u = y^3 - 3x^2y is a harmonic function.
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Evaluate ∫ (e^z + z) dz along the semi circle z = 2e^iθ, 0 ≤ θ ≤ π.
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Expand 1/(1+z^2) where |z| < 1.
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Evaluate ∫ (e^z / (z^2-z)) dz where C: |z|=2.
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Define cross ratio of four points.
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Prove that f(z) = z_bar is not conformal.
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Part B
Solve d^2y/dx^2 + x^2y = 0 by power series.
Answer
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Solve in series the equation d^2y/dx^2 + x dy/dx - x^2y = 0.
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Show that P_5(x) = (1/9)(5P_4(x) + 4P_2(x) + 0P_0(x)).
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Show that Σ n=0 to ∞ (P_n(x) z^n) = (1-2xz+z^2)^-1/2.
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If α and β are the roots of J_n(x) = 0 then prove the orthogonality property.
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If (u+v) = (sin 2x / cosh 2y - cos 2x) and f(z) = u + iv is an analytic function, find f(z).
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Prove that ∫ (sin^6 z / (z-π/6)^3) dz = 2πi (result).
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If F(a) = ∫ (z^2+3z+7)/(z-a) dz where c is |z|=2, find F(1), F(3), F'(1-i) and F''(1-i).
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Find the Taylor's and Laurent's series which represent (z-1)/(z+2)(z+3) in 2 < |z| < 3.
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Evaluate ∫ (e^z / (z^2+π^2)) dz where c: |z|=4 by residue theory.
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Evaluate ∫ (-∞ to ∞) (cos mx / (x^2+a^2)) dx (a>b>0).
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Show that the transformation w = (z+3)/(z-4) changes the circle x^2+y^2-4x=0 into the straight line 4u+3=0.
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Determine the region of the w-plane into which the region 1 ≤ x ≤ 2 and 1 ≤ y ≤ 2 is mapped by w = z^2.
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Show that the relation w = (5-4z)/(4z-2) transform the circle |z|=1 into a circle of radius unity.
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Find the bilinear transformation that maps the points 1, i, -1 into 2, i, -2.
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