Determine the nature of the point x = 0 for the equation (x2 + 1)y'' + (x2 - 1) + 2y = 0

Rewriting the equation as y'' + P(x)y' + Q(x)y = 0, we have P(x) = 0 and Q(x) = 2/(x2+1). Since P...

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JNTU B.Tech II MATHEMATICS-III Previous Question Paper (May/June - 2019 - R13)

B.Tech | II | I | R13 | May/June - 2019

Answer PaperPaper Code: 113AHMaximum Marks: 75Time: 3 Hours

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

125 marks

Determine the nature of the point x = 0 for the equation

(x² + 1)y'' + (x² - 1) + 2y = 0

2 marks

Answer

Rewriting the equation as y'' + P(x)y' + Q(x)y = 0, we have P(x) = 0 and Q(x) = 2/(x²+1). Since P(x) and Q(x) are analytic at x = 0, the point x = 0 is an ordinary point.

Find the indicial equation of

x²y'' - 2xy' - (x² - 2)y = 0

3 marks

Answer

For the equation x²y'' - 2xy' - (x² - 2)y = 0, we assume a Frobenius series solution y = Σ a_nx^n+r. Substituting this into the equation, the lowest power of x is x^r. The coefficient of x^r gives the indicial equation: r(r-1) - 2r + 2 = 0, which simplifies to r² - 3r + 2 = 0.

Write the value of

J_1/2(x)

2 marks

Answer

Obtain the value of

P₂(x)

3 marks

Answer

Write the Cauchy Riemann equations in polar form.

2 marks

Answer

Show that the function

f(z) = sin x cosh y + i cos x sinh y

is continuous and analytic everywhere.

3 marks

Answer

Define essential singularity.

2 marks

Answer

Expand

log z

by Taylor’s series about

z = 1

3 marks

Answer

Find the image of

z = 2 - i

under the transformation

w = z + 3 - i

2 marks

Answer

Prove that

w = 1/z

is circle preserving.

3 marks

Answer

Part B

Find the series solution of

4xy'' + 2y' + y = 0

10 marks

Answer

Solve the equation

3x(d²y/dx²) + (1 - x)(dy/dx) - y = 0

in power series.

10 marks

Answer

410 marks

Prove that

∫_-1¹ x P_n(x) P_n-1(x) dx = 2n / (4n² - 1)

5 marks

Answer

Show that

J₀² + 2(J₁² + J₂² + J₃² + ...) = 1

5 marks

Answer

If m₁, m₂ are roots of

J_n(x) = 0

, then prove that

∫₀¹ x J_n(m₁x) J_n(m₂x) dx = 0

10 marks

Answer

State and prove Cauchy’s Integral formula.

10 marks

Answer

Evaluate

∫_c (y - x - 3x²i) dz

, where c consists of the line segments from

z = 0

z = i

and the other from

z = i

z = 1 + i

10 marks

Answer

Evaluate

∫_-∞^∞ (z² - z + 2) / (z⁴ + 10z² + 9) dz

10 marks

Answer

Find Laurent expansion of

1 / (z² - 4z + 3)

for

1 < |z| < 3

10 marks

Answer

Determine the region of the w – plane into which the first quadrant of z – plane is mapped by the transformation

w = z²

10 marks

Answer

Show that every bilinear transformation maps the circles in the z – plane onto the circles in the w – plane.

10 marks

JNTU B.Tech II MATHEMATICS-III Previous Question Paper (May/June - 2019 - R13)

Part A

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Part B

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