MATHEMATICS—II (Differential Equations)

The solution of the differential equation $ (x^2y^2 + xy + 1)ydx + (x^2y^2 - xy + 1)xdy = 0 $ is $ px^2y^2 + qxy \log \left(\frac{cx}{y}\right) = 1 $, then the value of $ (p, q) $ is

The particular integral of the differential equation $ \frac{d^2y}{dx^2} - \frac{dy}{dx} - 6y = \sin 3x $ is $ \frac{1}{26}(p \cos 3x - q \sin 3x) $, then the value of $ (p, q) $ is

The particular integral of the partial differential equation $ xy \frac{\partial^2 z}{\partial x \partial y} = 1 $ is $ \log(ax) \log(by) $, then the value of $ (a, b) $ is

If $ z = ax + a^2 y^2 + b $ satisfies the partial differential equation $ \frac{\partial z}{\partial y} = c y \left(\frac{\partial z}{\partial x}\right)^d $, then the value of $ (c, d) $ is

$ \lim_{z \to 0} \frac{\bar{z}}{z} $ is

The sum of those poles which are lie in upper half plane of the function $ f(z) = \frac{1}{z^4 + a^4} $ is $ \frac{ia}{q} \sqrt{p} $, then the value of $ (p, q) $ is

The order of the pole of $ f(z) = \frac{\sin z}{z^4} $ is

If $ \Delta^n 0^m = [\Delta^n x^m]_{x=0} $ and $ \Delta^n 0^m = n [\Delta^{n-1} 0^{m-1} + \Delta^n 0^{m-1}] $, then the value of $ (p, q) $ is

The iterative formula by using the Newton-Raphson method for the real root of the equation $ x^2 + 4 \sin x = 0 $, is given by $ x_{n+1} = x_n - \frac{x^2 + p \sin x}{qx + 4 \cos x} $ then the value of $ (p, q) $ is

The order of the rate of convergence of Newton-Raphson method is

Solve :

$ \frac{dy}{dx} + y \cos x + y^2 \cos x \sin^2 x = 0 $

Solve :

$ (1 + y^2) + (x - e^{-\tan^{-1} y}) \frac{dy}{dx} = 0 $

Solve the following by the method of variation of parameters :

$ x^2 \frac{d^2y}{dx^2} - 2x(1 + x) \frac{dy}{dx} + 2(1 + x)y = x^3 $

Prove that

$ (2n + 1)x P_n = (n + 1) P_{n+1} + n P_{n-1} $

where $ P_n(x) $ is the Legendre's polynomial of the first kind.

Solve the partial differential equation

$ (D^2 - 2DD' + D'^2)z = 12xy $

Solve $ pxy + pq + qy = yz $, where $ p = \frac{\partial z}{\partial x} $ and $ q = \frac{\partial z}{\partial y} $

Find a bilinear transformation which maps points $ z_1, z_2, z_3 $ of the z-plane into points $ w_1, w_2, w_3 $ of the w-plane respectively.

Prove that $ u = e^{-x}(x \sin y - y \cos y) $ is harmonic.

Show that

$ \frac{1}{2\pi i} \oint_C \frac{e^{zt}}{z^2 + 1} dz = \sin t $

if $ t > 0 $ and C is the circle $ |z| = 3 $.

Using the theory of residues, evaluate the following integral, if $ a > 0 $, prove that

$ \int_0^\infty \frac{dx}{(x^2 + a^2)^2} = \frac{\pi}{4a^3} $

Find the deflection $ u(x, y, t) $ of the square membrane of side $ \pi $ and $ c^2 = 1 $ for initial velocity 0 and initial deflection $ \sin 2x \sin 4y $.

Find the electrostatic potential in the semi-disk $ r < 1, 0 < \theta < \pi $ which equals $ 110\theta(\pi - \theta) $ in the semicircle $ r = 1 $ and 0 on the segment $ -1 < x < 1 $.

The equation $ x^6 - x^4 - x^3 - 1 = 0 $ has one real root between (1.4, 1.5). Find this root to four decimal places by Regula-Falsi method.

Calculate $ \int_0^1 e^{\sin x} dx $ using Simpson's 3/8th rule by dividing the range into three equal parts.

Use Runge-Kutta method of 4th order to approximate y, when x = 1.1, given that $ y = 1.2 $ at x = 1 and $ \frac{dy}{dx} = 3x + y^2 $.

Apply Lagrange's formula to find the cubic polynomial which includes the following values of x and $ y_x $ :

Integrating factor of the differential equation $ dy = (e^{x-y})(e^x - e^y) dx $ is

The particular integral (PI) of the differential equation $ \frac{d^2 y}{dx^2} + 4y = \cos 2x $ is

The differential equation whose auxiliary equation has the roots 0, -1, -1 is

If $ J_0(x) $ and $ J_1(x) $ are Bessel functions, then $ J_1'(x) $ is given by

The complete solution of the partial differential equation $ p^2 + q^2 = 2 $ is

The partial differential equation $ 5 \frac{\partial^2 z}{\partial x^2} - 5 \left( \frac{\partial z}{\partial x} \right)^2 + 6 \frac{\partial^2 z}{\partial y^2} x = xy $ is classified as

The solution of the partial differential equation $ x^3 \frac{\partial u}{\partial x} + y^2 \frac{\partial u}{\partial y} = 0 $ if $ u(0, y) = 10 e^{5/y} $ using method of separation of variable is

If $ f(z) = u(x, y) + iv(x, y) $ is an analytic function, then $ f'(z) = $

The value of $ \int_C \frac{3z^2 + 7z + 1}{z + 1} dz $ where $ C $ is $ |z| = 1 $ is

The value of $ \Delta \tan^{-1}x $ is

Solve: $ 2ydx + x(2\log x - y)dy = 0 $

Solve: $ y - 2px = \tan^{-1}(xy^2) $

Use the method of variation of parameters to find the solution of the given differential equation $ y'' - 2y' + y = e^x \log x $

Find the series solution of the differential equation $ (x^2 - 2x + 1) \frac{d^2 y}{dx^2} + (4x - 4) \frac{dy}{dx} $
+ $ (x^2 - 2x + 3)y = 0 $ about the point $ x = 1 $.

Prove: $ \int J_3(x) dx = c - J_2(x) - \frac{2}{x} J_1(x) $, where $ c $ is arbitrary constant.

Prove: $ \int_{-1}^{1} x^2 P_{n+1}(x) P_{n-1}(x) dx = \frac{2n(n + 1)}{(2n - 1)(2n + 1)(2n + 3)} $

Solve: $ x^2(y - z) p + y^2(z - x) q = z^2(x - y) $

Find the complete integral of $ (p^2 + q^2)y = zq $

Prove that the general solution of the partial differential equation $ (D - mD' - a)^2 z = 0 $ where $ D = \frac{\partial}{\partial x}, D' = \frac{\partial}{\partial y} $ is given as $ z = e^{\alpha x} f_1(y + mx) + xe^{\alpha x} f_2(y + mx) $

Find the general solution of the given partial differential equation $ (D^3 D'^2 + D^3 D^2 - 5D^2 D'^2 - 2D^3 D' $ +
$ 6D^2 D')z = e^{x - y} $

Find the D'Alembert's solution of the wave equation $ \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} $

Find the solution of the differential equation $ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} $ subject to conditions—

(i) $ u $ is not infinite at $ t \to \infty $

(ii) $ \frac{\partial u}{\partial x} = 0 $ for $ x = 0 $ and $ x = \ell $

(iii) $ u = \ell x - x^2 $ for $ t = 0 $ between $ x = 0 $ and $ x = \ell $.

Describe the Runge-Kutta method of fourth order for the solution of initial value problem. Given the initial value problem $ y' = 1 + y^2 $, $ y(0) = 0 $. Find $ y(0.6) $ by using Runge-Kutta method of fourth order by taking $ h = 0.2 $.

Integrating factor of the differential equation $ dy = (e^{x-y})(e^x - e^y) dx $ is

The particular integral (PI) of the differential equation $ \frac{d^2 y}{dx^2} + 4y = \cos 2x $ is

The differential equation whose auxiliary equation has the roots 0, -1, -1 is

If $ J_0(x) $ and $ J_1(x) $ are Bessel functions, then $ J_1'(x) $ is given by

The complete solution of the partial differential equation $ p^2 + q^2 = 2 $ is

The partial differential equation $ 5 \frac{\partial^2 z}{\partial x^2} - 5 \left( \frac{\partial z}{\partial x} \right)^2 + 6 \frac{\partial^2 z}{\partial y^2} x = xy $ is classified as

The solution of the partial differential equation $ x^3 \frac{\partial u}{\partial x} + y^2 \frac{\partial u}{\partial y} = 0 $ if $ u(0, y) = 10 e^{5/y} $ using method of separation of variable is

If $ f(z) = u(x, y) + iv(x, y) $ is an analytic function, then $ f'(z) = $

The value of $ \int_C \frac{3z^2 + 7z + 1}{z + 1} dz $ where $ C $ is $ |z| = 1 $ is

The value of $ \Delta \tan^{-1}x $ is

Solve: $ 2ydx + x(2\log x - y)dy = 0 $

Solve: $ y - 2px = \tan^{-1}(xy^2) $

Use the method of variation of parameters to find the solution of the given differential equation $ y'' - 2y' + y = e^x \log x $

Find the series solution of the differential equation $ (x^2 - 2x + 1) \frac{d^2 y}{dx^2} + (4x - 4) \frac{dy}{dx} $
+ $ (x^2 - 2x + 3)y = 0 $ about the point $ x = 1 $.

Prove: $ \int J_3(x) dx = c - J_2(x) - \frac{2}{x} J_1(x) $, where $ c $ is arbitrary constant.

Prove: $ \int_{-1}^{1} x^2 P_{n+1}(x) P_{n-1}(x) dx = \frac{2n(n + 1)}{(2n - 1)(2n + 1)(2n + 3)} $

Solve: $ x^2(y - z) p + y^2(z - x) q = z^2(x - y) $

Find the complete integral of $ (p^2 + q^2)y = zq $

Prove that the general solution of the partial differential equation $ (D - mD' - a)^2 z = 0 $ where $ D = \frac{\partial}{\partial x}, D' = \frac{\partial}{\partial y} $ is given as $ z = e^{\alpha x} f_1(y + mx) + xe^{\alpha x} f_2(y + mx) $

Find the general solution of the given partial differential equation $ (D^3 D'^2 + D^3 D^2 - 5D^2 D'^2 - 2D^3 D' $ +
$ 6D^2 D')z = e^{x - y} $

Find the D'Alembert's solution of the wave equation $ \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} $

Find the solution of the differential equation $ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} $ subject to conditions—

(i) $ u $ is not infinite at $ t \to \infty $

(ii) $ \frac{\partial u}{\partial x} = 0 $ for $ x = 0 $ and $ x = \ell $

(iii) $ u = \ell x - x^2 $ for $ t = 0 $ between $ x = 0 $ and $ x = \ell $.

Describe the Runge-Kutta method of fourth order for the solution of initial value problem. Given the initial value problem $ y' = 1 + y^2 $, $ y(0) = 0 $. Find $ y(0.6) $ by using Runge-Kutta method of fourth order by taking $ h = 0.2 $.

Determine the type of the following differential equation:

$ \frac{d^2y}{dx^2} + \sin(x+y) = \sin x $

The singular solution of $ p = \log |px - y| $ is

If $ y = \phi(x) $ is a particular solution of $ y'' + (\sin x)y' + 2y = e^x $ and $ y = \psi(x) $ is a particular solution of $ y'' + (\sin x)y' + 2y = \cos 2x $, then particular solution of $ y'' + (\sin x)y' + 2y = e^x + 2 \sin^2 x $ is given by

Let $ D = \frac{d}{dx} $. Then the value of $ [1 / (xD + 1)]x^{-1} $ is

If $ P_n $ is the Legendre polynomial of first kind, then the value of $ \int_{-1}^1 x P_n P_n' dx $ is

The partial differential equation $ y^3 u_{xx} - (x^2 - 1) u_{yy} = 0 $ is

If $ u(x, t) $ satisfies the partial differential equation $ \frac{\partial^2u}{\partial t^2} - 4 \frac{\partial^2u}{\partial x^2} = 0 $, then $ u(x, t) $ can be of the form

If $ J_n $ is the Bessel's function of first kind, then the value of $ J_{1/2} $ is

The region in which the following partial differential equation $ x^2 \frac{\partial^2u}{\partial x^2} + 2y \frac{\partial^2u}{\partial y^2} - 3 \frac{\partial^2u}{\partial x\partial y} + 5u = 0 $ acts as parabolic equation is

$ \frac{d^2y}{dx^2} = 6x + 2 $, $ y(0)=0 $, $ y(1)=0 $. The value of $ \frac{d^2y}{dx^2} $ at $ x = 1/2 $ using the finite difference method and a step size of $ h = 1/4 $ can be approximated by

Solve $ (xy^2 + e^{-1/x^3})dx - x^2ydy = 0 $

Solve $ p^2 + 2py \cot x = y^2 $

Solve the following differential equation :

$ e^{3x}(p-1) + p^3 e^{2y} = 0 $

Use the method of variation of parameters to find the solution of the given differential equation :

$ y'' - 3y' + 2y = \cos(e^{-x}) $

Express $ x^4 + 2x^3 + 2x^2 - x - 3 $ in terms of Legendre's polynomials.

Prove that

$ 1 + \sum_{n=1}^\infty \frac{1}{n+1} P_n(\cos \theta) = \log(1 + \csc(\theta/2)) $

Solve the following differential equation :

$ (y+z)p + (z+x)q = x+y $

Find the complete integral of $ z^2 p^2 y + 6zpxy + 2zqx^2 + 4x^2y = 0 $

Find the solution of following partial differential equation :

$ r - t = \tan^3 x \tan y - \tan x \tan^3 y $

Solve: $ (2D - 3D' + 7)^2 (D^2 + 3D') z = 0 $

Classify the partial differential equation :

$ \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + x \frac{\partial^2u}{\partial x\partial y} + 2 \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + 6u = 0 $

Find the solution of Laplace's equation in cylindrical coordinates.

The points of trisection of a string are pulled aside through the same distance on opposite sides of the position of equilibrium and the string is released from rest. Derive an expression for the displacement of the string at subsequent time and show that the midpoint of the string always remains at rest.

Solve the initial value problem $ yy' = x $, $ y(0)=1 $, using the Euler method in $ 0 \le x \le 0.8 $, with $ h=0.2 $ and $ h=0.1 $. Compare the results with the exact solution at $ x=0.8 $. Extrapolate the result.

Write the bound on the truncation error of the Taylor series method.