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 \(x^{2} p+y^{2} q+z^{2}=0, p \equiv \dfrac{\partial z}{\partial x}, q \equiv \dfrac{\partial z}{\partial y}\) which passes through the hyperbola \(xy = x + y\), \(z =1\).

[20M]

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

[20M]

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

[20M]

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

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

[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 \(2 x z- px ^{2}-2 qxy + pq =0\).

[20M]

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

[20M]

6) Find the general solution of \(x^{2} r-y^{2} t+x p-y q=\log x\)

[20M]

1992

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

[20M]

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

[20M]

3) Use Charpit’s method to solve \(p x+q y=z \sqrt{1+p q}\)

[20M]

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

[20M]

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

[20M]

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

[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 \(p^{3}+q^{3}=3 p q z\).

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

4) Use Charpit’s method to solve \(2 z x-p x^{2}-2 q x y+p q=0\).

5) Solve the homogeneous liner differential equation \(\dfrac{\partial^{2} z}{\partial x^{2}}+5 \dfrac{\partial^{2} z}{\partial x \partial y}+6 \dfrac{\partial^{2} z}{\partial y^{2}}=\dfrac{1}{y-2 x}\).

6) Find the complementary function and particular integral of \(\dfrac{\partial^{2} z}{\partial x \partial y}+\dfrac{\partial z}{\partial x}-\dfrac{\partial z}{\partial y}=z+x y\).

1990

1) Solve by Charpit’s method

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

3) Solve completely the equation \(z=p x+q y+\dfrac{q}{p}-p\) and classify the following integrals of this equation, \(z=2 x+4 y\), \(y z=1-x\), \(x^{2}+4 y z=0\).

4) Show that the general solution of \(x p-y q=2 x e^{-\left(x^{2}+y^{2}\right)}\) can be expressed in the form

5) Solve \(p y+q x+p q=0\).

6) Find the general solution \(p\left(y^{2}+z^{2}\right)-q x y+x z=0\).

1989

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

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

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

4) Solve completely \(p q=x^{\prime \prime \prime} y^{\prime \prime} z^{\prime}\).

5) Solve \(z=p x+q y+\sqrt{1+p^{2}+q^{2}}\). Find singular solution.

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