Mathematics - II (Differential Equation)

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?

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

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

The solution of $ p+q=z $ is

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

The transformation $ w = \sin z $ is conformal

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

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

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

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

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.

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?

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

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

The solution of $ p+q=z $ is

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

The transformation $ w = \sin z $ is conformal

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

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

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

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

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.