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 $x{p}^2-ypq+y^{2}q-y^{2}z=0$.

3) Solve $(y^2+z^2)p-yxq=-zx$.

4) Solve completely $z=px+qy+3p^{{\displaystyle \frac{1}{3}}}{q}^{{\displaystyle \frac{1}{3}}}$.

5) Solve ${\displaystyle \frac{x^2}{p}}+{\displaystyle \frac{y^2}{q}}=z$.

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

1987

1) Solve

(i) $x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}-y^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=xy$
(ii) $s-t=x{y}^{-2}$
(iii) $r=a^{2}t$

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

$y(0,\mathrm{t})=0,\mathrm{y}=(l,\mathrm{t})=\mathrm{A}\sin wt$, ${\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }t}}=0$ 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) $p^2+q^2-2px-2qy+1=0$.
(ii) ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}+3{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}=x+y$.

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

1985

1) Solve the differential equation

2) Reduce the equation $y^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}-2xy{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}={\displaystyle \frac{y^2}{y}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}}+{\displaystyle \frac{x^2}{x}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}$ to canonical form and hence solve it.

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

when $t=0$

TBC

1984

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

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

3) Solve $x{q}^{2}r-2xpqs+x{p}^{2}t+2p{q}^2=0$.

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

1983

1) Find the complete and singular integral of the differential equation $z=xp+yq+p^2-q^2$, 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 $pq+x(2y+1)p+y(y+1)q-(2y+1)z=0$.

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

4) Find the solution of the heat equation ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}=c^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$ in the case of a semi-infinite bar extending from 0 to $\mathrm{\infty },$ 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

$u(x,t)={\displaystyle \frac{1}{\sqrt{\pi }}}[{\int }_{-\pi /\tau }^{\mathrm{\infty }}f(x+\tau w)e^{-w^2}dw-{\int }_{x/\tau }^{\mathrm{\infty }}f(-x+\tau w)e^{-w^2}dw]$,

where $\tau =2c\sqrt{t}$