MATHEMATICS—II (Differential Equations) - B.Tech 2nd Semester Exam., 2021

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

1. \( e^{e^x} \)
2. \( e \)
3. \( e^x \)
4. \( e^{2x} \)

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

1. \( \frac{\sin 2x}{2} \)
2. \( \frac{x \sin 2x}{2} \)
3. \( \frac{x \sin 2x}{4} \)
4. \( \frac{x \cos 2x}{2} \)

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

1. \( \frac{d^3 y}{dx^3} - 2\frac{d^2 y}{dx^2} + \frac{dy}{dx} = 0 \)
2. \( \frac{d^3 y}{dx^3} + 2\frac{d^2 y}{dx^2} + \frac{dy}{dx} = 0 \)
3. \( \frac{d^2 y}{dx^2} + 2\frac{dy}{dx} + y = 0 \)
4. \( \frac{d^3 y}{dx^3} + 2\frac{d^2 y}{dx^2} + y = 0 \)

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

1. \( -J_0(x) \)
2. \( J_0(x) - \frac{1}{x} J_1(x) \)
3. \( J_0(x) + \frac{1}{x} J_1(x) \)
4. \( J_0(x) \)

The complete solution of the partial differential equation \( p^2 + q^2 = 2 \) is

1. \( z = ax + \sqrt{2 - a^2} y + c \)
2. \( z = ax - \sqrt{2 - a^2} y + c \)
3. Either (i) or (ii) true
4. (i) and (ii) both true

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

1. parabolic
2. hyperbolic
3. elliptic
4. None of the above

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

1. \( 10 e^{5/2x^2} e^{5/y} \)
2. \( 10 e^{-5/2y^2} e^{5/x} \)
3. \( 10 e^{-5/2y^2} e^{-5/x} \)
4. \( 10 e^{-5/2x^2} e^{5/y} \)

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

1. \( \frac{\partial u}{\partial x} - i \frac{\partial v}{\partial x} \)
2. \( \frac{\partial u}{\partial x} - i \frac{\partial u}{\partial y} \)
3. \( \frac{\partial v}{\partial y} - i \frac{\partial v}{\partial x} \)
4. \( \frac{\partial u}{\partial x} + i \frac{\partial u}{\partial y} \)

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

1. 2πi
2. 0
3. πi
4. πi/2

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

1. \( \Delta \tan^{-1} \left( \frac{h}{1 + hx + x^2} \right) \)
2. \( \Delta \tan^{-1} \left( \frac{h + 2x}{1 + hx + x^2} \right) \)
3. \( \Delta \tan^{-1} \left( \frac{h - 2x}{1 + hx + x^2} \right) \)
4. \( \Delta \tan^{-1} \left( \frac{h + 2x}{1 - hx - x^2} \right) \)

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 \( h = 0.2 \).