PYQs 5

We will cover following topics

2004
2003
2002
2001
2000

2004

1) Find the general solution of the partial differential equation $(z^{2} - 2 y z - y^{2}) p + x (y + z) q = x (y - z)$

2) Apply charpit’s method to find the complete integral of the partial differential equation $p x y + p q + q y = y z$ .

3) Solve the Laplace equation $\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0, 0 \leq x \leq π, 0 \leq y \leq \infty$ satisfying the boundary conditions $u (0, y) = 0,$ for $0 \leq y < \infty$ $u (π, y) = 0,$ for $0 < y < \infty$ $u (x, \infty) = 0,$ for $0 < x < π$ and $u (x, 0) = u_{0},$ for $0 < x < π$

2003

1) Find the general solution of the partial differential equation $(m z - n y) \frac{\partial z}{\partial x} + (n x - l z) \frac{\partial z}{\partial y} = l y - m x$ .

2) Form the partial differential equation by eliminating the arbitrary function from $f (x^{2} + y^{2}, z - x y) = 0, z = z (z, y)$ .

3) Solve $\frac{\partial u}{\partial t} = c^{2} \frac{\partial^{2} u}{\partial t^{2}}$ given that (i) $u = 0,$ When $t = 0 t = 0$ for all $t$ (ii) $u = 0,$ When $x = l$ for all $t$ (iii) $\begin{aligned} u & = x in (0, l / 2) \\ = l - x in (l / 2, l) \end{aligned}} at t = 0$

2002

1) Solve completely $\frac{x}{p q} = \frac{x}{q} + \frac{y}{p} + \sqrt{p q}$ .

2) Using charpit’s method solve completely $p^{2} - q^{2} = (x + y)^{2}$ .

3) Obtain the general solution of the following equation $\frac{\partial^{2} z}{\partial x^{2}} - 3 \frac{\partial^{2} z}{\partial x \partial y} + 2 \frac{\partial^{2} z}{\partial y^{2}} = (2 + 4 x) e^{x - 2 y}$ .

2001

1) Find the complete integral of the partial differential equation $x^{2} p^{2} + y^{2} q^{2} = z^{2}$ .

2) Solve by Charpit’s method $(p^{2} + q^{2}) y = q z$ .

3) If $φ (x)$ is a continuous and bounded function for $- \infty < x < \infty,$ prove that the function $u (x, t) = \frac{1}{2 \sqrt{π k t}} \int_{- \infty}^{\infty} φ (ξ) e^{- (x - ξ)^{2} / 4 x} d ξ$ is a solution of the initial value problem: $u_{t} - k u_{x x} = 0, - \infty < x < \infty, t > 0$ $u (x, 0) = φ (x)$ for $- \infty < x < \infty$ .

2000

1) Solve the following initial value problem $(y + z) z_{x} + y z_{y} = x - y; z = 1 + t$ on die initial curve $C : x = t, y = 1; - \infty < T < \infty$ .

2) Determine the complete integral of the equation $z \frac{\partial z}{\partial x} \frac{\partial z}{\partial y} = {(\frac{\partial z}{\partial x})}^{2} (x \frac{\partial z}{\partial y} + {[\frac{\partial z}{\partial x}]}^{2}) + {(\frac{\partial z}{\partial y})}^{2} (y \frac{\partial z}{\partial x} + {[\frac{\partial z}{\partial y}]}^{2})$ .

3) Solve the following partial differential equation $\frac{\partial^{2} z}{\partial x^{2}} + \frac{\partial^{2} z}{\partial x \partial y} - 2 \frac{\partial^{2} z}{\partial y^{2}} - \frac{\partial z}{\partial x} - 2 \frac{\partial z}{\partial y} = 0$ .

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