1988

1) Describe the Charpit’s method of solving the equation \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{p}, \mathrm{q})=0\).

2) Solve \(x p^{2}-y p q+y^{2} q-y^{2} z=0\).

3) Solve \(\left(y^{2}+z^{2}\right) p-y x q=-z x\).

4) Solve completely \(z=p x+q y+3 p^{\dfrac{1}{3}} q^{\dfrac{1}{3}}\).

5) Solve \(\dfrac{x^{2}}{p}+\dfrac{y^{2}}{q}=z\).

6) Solve \(9\left(p^{2} z+q^{2}\right)=4\).

1987

1) Solve

(i) \(x^{2} \dfrac{\partial^{2} z}{\partial x^{2}}-y^{2} \dfrac{\partial^{2} z}{\partial y^{2}}=x y\)
(ii) \(s-t=x y^{-2}\)
(iii) \(r=a^{2} t\)

2) Solve the boundary value problem \(\dfrac{\partial^{2} y(x, t)}{\partial t^{2}}=a^{2} \dfrac{\partial^{2} y(x, t)}{\partial x^{2}}\) under the boundary conditions:

\(y(0, \mathrm{t})=0, \mathrm{y}=(l, \mathrm{t}) = \mathrm{A} \sin wt\), \(\dfrac{\partial y}{\partial t}=0\) at \(t=0\), \(y(x, 0)=0\) at \(t=0\).

3) Find the function \(\mathrm{u}(\mathrm{x}, \mathrm{y})\) which satisfies the Laplace’s equation in the rectangle \(0<\mathrm{x}<\mathrm{a}, 0<\mathrm{y}<\mathrm{b}\), and which also satisfies the boundary conditions \(u_{y}(x, 0)=0\), \(u_{y}(x, b)=0\), \(u_{x}(0, y)=0\), \(u_{x}(a, y)=f(y)\).

1986

1) Solve (i) \(p^{2}+q^{2}-2 p x-2 q y+1=0\).
(ii) \(\dfrac{\partial^{2} z}{\partial x^{2}}+3 \dfrac{\partial^{2} z}{\partial x \partial y}+2 \dfrac{\partial^{2} z}{\partial y^{2}}=x+y\).

2) Solve \(\dfrac{\partial^{2} y}{\partial x^{2}}=\dfrac{1}{c^{2}} \dfrac{\partial^{2} y}{\partial t^{2}}\) under the boundary conditions:
\(\mathrm{y}=0\) for \(\mathrm{x}=0\) and for all values of \(t\)
\(\dfrac{\partial y}{\partial t}=0\) for \(\mathrm{t}=0\) and for all values of \(\mathrm{x}\)
\(\mathrm{y}=0\) for \(\mathrm{x}=\pi\) and for all values of \(\mathrm{t}\)
\(y=\dfrac{h x}{d}\) for \(0^{\prime \prime} x^{\prime \prime} d\) and \(t=0\) and
\(y=\dfrac{h(l-x)}{(l-d)}\) for \(d \leq x \leq \pi\) and \(t=0\).

1985

1) Solve the differential equation

2) Reduce the equation \(y^{2} \dfrac{\partial^{2} z}{\partial x^{2}}-2 x y \dfrac{\partial^{2} z}{\partial x \partial y}+x^{2} \dfrac{\partial^{2} z}{\partial y^{2}}=\dfrac{y^{2}}{y} \dfrac{\partial z}{\partial x}+\dfrac{x^{2}}{x} \dfrac{\partial z}{\partial y}\) to canonical form and hence solve it.

3) Find a solution of \(\dfrac{\partial^{2} y}{\partial t^{2}}=c^{2} \dfrac{\partial^{2} y}{\partial x^{2}}\) such that:
(i) $y$ involves a trigonometrically.
(ii) \(\mathrm{y}=0\) when \(\mathrm{x}=0\) or \(\pi,\) for all values of \(\mathrm{t}\)
(iii) \(\dfrac{\partial y}{\partial t}=0\) when \(\mathrm{t}=0\) for all values of \(\mathrm{x}\)
(iv) \(\left.\begin{array}{l}y=\sin x \text { from } x=0 \text { to } x=\dfrac{\pi}{2} \\ y=0 \text { from } x=\dfrac{\pi}{2} \text { to } x=\pi\end{array}\right\}\)

when \(t=0\)

TBC

1984

1) Find a complete integral of the equation \(2 z q^{2}-y^{2} p+y^{2} q=0\).

2) Solve the equation \(\dfrac{\partial^{2} z}{\partial x^{2}}-3 \dfrac{\partial^{2} z}{\partial x \partial y}+2 \dfrac{\partial^{2} z}{\partial y^{2}}-\dfrac{\partial z}{\partial x}+2 \dfrac{\partial z}{\partial y}\)=\((2+4 x) e^{-y}\).

3) Solve \(x q^{2} r-2 x p q s+x p^{2} t+2 p q^{2}=0\).

4) Solve Laplace’s equation \(\rho^{2} \dfrac{\partial^{2} y}{\partial \rho^{2}}+\rho \dfrac{\partial y}{\partial \rho}+\dfrac{\partial^{2} y}{\partial \theta^{2}}=0\) under the boundary condition. \(y(\rho, 0)=0,0 \leq \rho<10\)
\(y(\rho, \pi)=0,0<\rho<10\) \(y(10, \theta)=\dfrac{200 \theta}{\pi}, 0 \leq \theta<\dfrac{\pi}{2}\)
\(y(10, \theta)=\dfrac{200}{\pi}(\pi-\theta), \dfrac{\pi}{2}<\theta<\pi\)

1983

1) Find the complete and singular integral of the differential equation \(z=x p+y q+p^{2}-q^{2}\), find also a developable surface belonging to the general integral of this differential equation.

2) Find the complete and singular integral of the differential equation \(p q+x(2 y+1) p+y(y+1) q-(2 y+1) z=0\).

3) Solve \(\dfrac{\partial^{2} y}{\partial t^{2}}=a^{2} \dfrac{\partial^{2} y}{\partial x^{2}}\) under the boundary conditions: \(y(0, t)=0, t>0\); \(y(L, t)=0, t>0\) and the initial conditions \(y(x, 0)=m\left(L x-x^{2}\right), 0<x<L\) and \(\left(\dfrac{\partial y}{\partial t}\right)_{t=0}=0\) \(0<x<\mathrm{L}\) where \(\mathrm{m}\) is a suitable constant.

4) Find the solution of the heat equation \(\dfrac{\partial u}{\partial t}=c^{2} \dfrac{\partial^{2} u}{\partial x^{2}}\) in the case of a semi-infinite bar extending from 0 to \(\infty,\) the end at \(x=0\) is held at temperature zero and the initial temperature is \(f(x)\). Show that the solution may be written as

\(u(x, t)=\dfrac{1}{\sqrt{\pi}}\left[\int_{-\pi / \tau}^{\infty} f(x+\tau w) e^{-w^{2}} d w-\int_{x / \tau}^{\infty} f(-x+\tau w) e^{-w^{2}} d w\right]\),

where \(\tau=2 c \sqrt{t}\)