JNTU
JNTU B.Tech II MATHEMATICS-III Previous Question Paper (February - 2024 - R15)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Define Cauchy’s linear differential equation.
Answer
A Cauchy’s linear differential equation (also known as a Cauchy-Euler equation) is a linear differential equation of the form:
where an, an-1, ..., a0 are constants and Q(x) is a function of x. The defining characteristic is that the power of the independent variable x in each term matches the order of the derivative in that same term.
What are the ordinary point and regular singular points of a differential equation?
Answer
For a second-order linear differential equation of the form
- Ordinary Point: The point x₀ is an ordinary point if both P(x) and Q(x) are analytic at x₀, meaning they possess a Taylor series expansion about that point.
- Regular Singular Point: The point x₀ is a singular point if it is not an ordinary point. It is classified as a regular singular point if the functions (x - x₀)P(x) and (x - x₀)²Q(x) are both analytic at x₀.
State Legendre’s equation.
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Prove that J_1(x) = -J'_0(x).
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Define an analytic function and give an example.
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State Cauchy’s generalized integral formula. What is the use of it?
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State the Maclaurin’s series expansion of f(z) in a complex plane.
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Define a pole of order n. Give a suitable example.
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What is the condition for the existence of conformal mapping?
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Discuss the transformation w = z + c, where c is a complex constant.
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Part B
Solve (1+x^2) d^2y/dx^2 + (1+x) dy/dx + y = sin(2 log(1+x)).
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Solve in series the equation d^2y/dx^2 + xy = 0.
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Prove that J_4(x) = (48/x^2 - 1)J_2(x) - (24/x - x/4)J_1(x).
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State and prove orthogonal property of Legendre’s polynomials.
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Prove that the function f(z) defined by f(z) = (x^3(1+i) - y^3(1-i)) / (x^2+y^2) for z is not 0, and f(0)=0 is continuous and the Cauchy-Riemann equations are satisfied at the origin, yet f'(0) does not exist.
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Evaluate integral of e^z / (z^2+1)^2 dz around C: |z-i|=3.
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Find the Laurent’s series expansion of (z-7) / ((z+1)(z-2)) in the annulus 1 < |z+1| < 3.
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Find the residue of f(z) = z^3 / ((z-1)(z-2)(z-3)) at its poles and hence evaluate the integral over the circle |z|=2.5.
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Find the image of the rectangle -pi <= x <= pi, -1/2 <= y <= 1/2 under the mapping w = sin z.
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Find the bilinear transformation which maps the points z = -1, i, 1 onto the points w = -i, 0, i. Hence find (a) the image of |z|<1 and (b) the invariant points of the transformation.
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