Mathematics - II (Differential Equation) - B.Tech 2nd Semester Examination, 2025 (Old Course)

For the differential equation, \( (x-1) \frac{d^2y}{dx^2} + \cot(\pi x) \frac{dy}{dx} + ( cosec^2 \pi x)y = 0 \) which of the following statement is true?

1. 0 is regular and 1 is irregular
2. 0 is irregular and 1 is regular
3. Both 0 and 1 are regular
4. Both 0 and 1 are irregular

Let \( P_n(x) \) be the Legendre polynomial of degree \( n \ge 0 \). Then \( \int_{-1}^{1} P_n(x) dx = 2 \) if n is

1. 0
2. 1
3. 2
4. None of these

\( \int_{0}^{\pi/2} \sqrt{\pi x} J_{1/2}(2x) dx = k \) then k is

1. 0
2. 1
3. 2
4. \( \sqrt{\pi} \)

The solution of \( p+q=z \) is

1. \( f(xy, y \log z) = 0 \)
2. \( f(x+y, y+\log z) = 0 \)
3. \( f(x-y, y-\log z) = 0 \)
4. \( f(x/y, y/\log z) = 0 \)

The complementary function of \( \frac{\partial^2 z}{\partial x^2} - 4 \frac{\partial^2 z}{\partial x \partial y} + 4 \frac{\partial^2 z}{\partial y^2} = x+y \) is

1. \( f(y+2x) + xg(y+2x) \)
2. \( f(y-2x) + xg(y-2x) \)
3. \( x f(y+2x) + y g(y+2x) \)
4. \( x f(y+2x) - y g(y+2x) \)

The transformation \( w = \sin z \) is conformal

1. at all points
2. at all points except \( z = n\pi + \frac{\pi}{2} \), \( n=0, \pm 1, \pm 2, \dots \)
3. at all points except \( z = n\pi + \frac{\pi}{2} \), \( n=0, 1, 2, \dots \)
4. at all points except \( z = n\pi + \frac{\pi}{2} \), \( n=1, 2, 3, \dots \)

The value of \( \int_C \frac{3z^2 + 7z + 1}{z-4} dz \), where C is \( 9x^2 + 4y^2 = 36 \) is

1. \( 2\pi i \)
2. 0
3. \( 3\pi i \)
4. \( 4\pi i \)

Newton's iterative formula to find the value of \( \sqrt{N} \) is

1. \( x_{n+1} = \frac{1}{2} (x_n - N/x_n) \)
2. \( x_{n+1} = \frac{1}{2} (x_n - Nx_n) \)
3. \( x_{n+1} = \frac{1}{2} (x_n + N/x_n) \)
4. \( x_{n+1} = \frac{1}{2} (x_n + Nx_n) \)

While applying Simpson's 3/8 rule the number of sub intervals should be

1. Odd
2. 8
3. Even
4. Multiple of 3

In the geometrical meaning of Euler's algorithm, the curve is approximated as a

1. Straight line
2. Circle
3. Parabola
4. Ellipse

Solve \( y = 2px + y^2 p^3 \).

Solve \( (px-y)(py+x) = a^2 p \).

Find the solution of the differential equation \( x^2 y'' - 4xy' + 6y = x^4 \cos x \), \( y(\pi)=0, y'(\pi)=1 \).

State and prove the orthogonal property of Legendre polynomial.

Find the solution of the following partial differential equation: \( (x^2-yz)p + (y^2-zx)q = z^2-xy \)

Find the solution of the following partial differential equation: \( 2z + p^2 + qy + 2y^2 = 0 \)

Find the solution of the partial differential equation \( (D - 5D' - 6)z = 6e^{6x} \log(y+5x) \).

Show that the function \( f(z) = \sqrt{|xy|} \) is not analytic at the origin even though C.R. equations are satisfied thereof.

Find the Taylor series expansion of the function \( f(z) = \sin z \). Also find the values of \( f^{(2n)}(0) \) and \( f^{(2n-1)}(0) \).

Write the statement of Cauchy Integral formula. Hence, evaluate the integral \( \int_C \frac{z-1}{(z+1)^2(z-2)} dz \) where C is the circle \( |z-i|=2 \).

Find the value of the integral \( \int_{0}^{\infty} \frac{\cos ax - \cos bx}{x^2} dx \) where \( a, b > 0 \).

Discuss the rate of convergence of Bisection method.

A third degree polynomial passes through the points (0,1), (1, 1), (2, 1) and (3, -2). Find the polynomial.

Evaluate \( \int_{4}^{5/2} \log x \, dx \) by Simpson's 1/3 rule and Simpson's 3/8 rule, by dividing the range into 6 parts.

Using Euler's modified method, find numerical solution of the differential equation \( \frac{dy}{dx} = x + |\sqrt{y}| \) with \( y(0)=1 \) for \( 0 \le x \le 0.6 \), in steps of 0.2.