1994

1) Find the differential equation of the family of all cones with vertex at $(2,-3,1)$.

[20M]

2) Find the integral surface of $x^{2}p+y^{2}q+z^2=0,p\equiv {\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}},q\equiv {\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}$ which passes through the hyperbola $xy=x+y$, $z=1$.

[20M]

3) Use Charpit’s method to solve $16p^2{z}^2+9q^2{z}^2+4z^2-4=0$ Interpret geometrically the complete solution and mention the singular solution.

[20M]

4) Solve $(D^2+3D{D}^{\mathrm{\prime }}+2D^2)z=x+y,$ by expanding the particular integral in ascending powers of $D,$ as well as in ascending powers of $D$ ‘

[20M]

5) Find a surface satisfying $(D^2+D{D}^{\mathrm{\prime }})z=0$ and touching the elliptic paraboloid $z=4x^2+y^2$ along its section by the plane $y=2x+1$.

[20M]

1993

1) Find the surface whose tangent planes cut off an intercept of constant length $R$ from the axis of $z$.

[20M]

2) Solve $(x^2+3x{y}^2)p+(y^3+3x^{2}y)q=2(x^2+y^2)z$

[20M]

3) Find the integral surface of the partial differential equation $(x-y)p+(y-x-z)q=z$ through the circle $z=1$, $x^2+y^2=1$.

[20M]

4) Using Charpit’s method find the complete integral of $2xz-p{x}^2-2qxy+pq=0$.

[20M]

5) Solve $r-s+2q-z=x^2{y}^2$.

[20M]

6) Find the general solution of $x^{2}r-y^{2}t+xp-yq=\log x$

[20M]

1992

1) Solve: $(2x^2-y^2+z^2-2yz-zx-xy)p+(x^2+2y^2+z^2-yz-2zx-xy)q=x^2+y^2+2z^2-yz-zx-2xy$

[20M]

2) Find the complete integral of $(y-x)(qy-px)=(p-q{)}^2$

[20M]

3) Use Charpit’s method to solve $px+qy=z\sqrt{1+pq}$

[20M]

4) Find the surface passing through the parabolas $\begin{array}{l}z=0,y^2=4ax\\ z=1,y^2=-4ax\end{array}$ and satisfying the differential equation $xr+2p=0$

[20M]

5) Solve:
$r+s-6t=y\cos x$

[20M]

6) Solve:
${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}-z=\cos (x+2y)+e^y$

[20M]

1991

1) Explain the terms complete integral, particular integral, general integral and singular integral with reference to a partial differential equation of the first order in two independent variables.

2) Solve $p^3+q^3=3pqz$.

3) Solve $(x+y)(p+q{)}^2+(x-y)(p-q{)}^2=1$.

4) Use Charpit’s method to solve $2zx-p{x}^2-2qxy+pq=0$.

5) Solve the homogeneous liner differential equation ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}+5{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+6{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{y}^2}}={\displaystyle \frac{1}{y-2x}}$.

6) Find the complementary function and particular integral of ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }x\mathrm{\partial }y}}+{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}}-{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}=z+xy$.

1990

1) Solve by Charpit’s method

2) Find the complete integral of ${\left({\displaystyle \frac{x}{p}}\right)}^n+{\left({\displaystyle \frac{y}{q}}\right)}^n=z^n$.

3) Solve completely the equation $z=px+qy+{\displaystyle \frac{q}{p}}-p$ and classify the following integrals of this equation, $z=2x+4y$, $yz=1-x$, $x^2+4yz=0$.

4) Show that the general solution of $xp-yq=2x{e}^{-(x^2+y^2)}$ can be expressed in the form

5) Solve $py+qx+pq=0$.

6) Find the general solution $p(y^2+z^2)-qxy+xz=0$.

1989

1) Using Charpit’s method solve the equation $zp(x+y)+q(q-p)-z^2=0$.

2) Show how to solve the equation $\mathrm{P}p+\mathrm{Q}q=\mathrm{R}$, where $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ are functions of $\mathrm{x}$, $\mathrm{y}$, $\mathrm{z}$.

3) Show that the integral of $\varphi \left({\displaystyle \frac{y}{x}}\right)p+\psi \left({\displaystyle \frac{y}{x}}\right)q=1$ can be obtained as $z={\displaystyle \frac{x}{I}}\int {\displaystyle \frac{1}{\psi }}{\displaystyle \frac{dI}{dv}}dv+F\left({\displaystyle \frac{x}{I}}\right)$, where $v={\displaystyle \frac{y}{x}}$, $\log I=\int {\displaystyle \frac{\varphi dv}{\psi -v\varphi }}$ and $\mathrm{F}$ is arbitrary.

4) Solve completely $pq=x^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}{y}^{\mathrm{\prime }\mathrm{\prime }}{z}^{\mathrm{\prime }}$.

5) Solve $z=px+qy+\sqrt{1+p^2+q^2}$. Find singular solution.

6) Solve $z^2(p^2+q^2)=x^2+y^2$.