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 x2p+y2q+z2=0,p≡∂z∂x,q≡∂z∂y which passes through the hyperbola xy=x+y, z=1.

[20M]

3) Use Charpit’s method to solve 16p2z2+9q2z2+4z2−4=0 Interpret geometrically the complete solution and mention the singular solution.

[20M]

4) Solve (D2+3DD′+2D2)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 (D2+DD′)z=0 and touching the elliptic paraboloid z=4x2+y2 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 (x2+3xy2)p+(y3+3x2y)q=2(x2+y2)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, x2+y2=1.

[20M]

4) Using Charpit’s method find the complete integral of 2xz−px2−2qxy+pq=0.

[20M]

5) Solve r−s+2q−z=x2y2.

[20M]

6) Find the general solution of x2r−y2t+xp−yq=logx

[20M]

1992

1) Solve: (2x2−y2+z2−2yz−zx−xy)p+(x2+2y2+z2−yz−2zx−xy)q=x2+y2+2z2−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√1+pq

[20M]

4) Find the surface passing through the parabolas z=0,y2=4axz=1,y2=−4ax and satisfying the differential equation xr+2p=0

[20M]

5) Solve:
r+s−6t=ycosx

[20M]

6) Solve:
∂2u∂x2∂2z∂x∂y+∂z∂y−z=cos(x+2y)+ey

[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 p3+q3=3pqz.

3) Solve (x+y)(p+q)2+(x−y)(p−q)2=1.

4) Use Charpit’s method to solve 2zx−px2−2qxy+pq=0.

5) Solve the homogeneous liner differential equation ∂2z∂x2+5∂2z∂x∂y+6∂2z∂y2=1y−2x.

6) Find the complementary function and particular integral of ∂2z∂x∂y+∂z∂x−∂z∂y=z+xy.

1990

1) Solve by Charpit’s method

2) Find the complete integral of (xp)n+(yq)n=zn.

3) Solve completely the equation z=px+qy+qp−p and classify the following integrals of this equation, z=2x+4y, yz=1−x, x2+4yz=0.

4) Show that the general solution of xp−yq=2xe−(x2+y2) can be expressed in the form

5) Solve py+qx+pq=0.

6) Find the general solution p(y2+z2)−qxy+xz=0.

1989

1) Using Charpit’s method solve the equation zp(x+y)+q(q−p)−z2=0.

2) Show how to solve the equation Pp+Qq=R, where P, Q, R are functions of x, y, z.

3) Show that the integral of ϕ(yx)p+ψ(yx)q=1 can be obtained as z=xI∫1ψdIdvdv+F(xI), where v=yx, logI=∫ϕdvψ−vϕ and F is arbitrary.

4) Solve completely pq=x′′′y′′z′.

5) Solve z=px+qy+√1+p2+q2. Find singular solution.

6) Solve z2(p2+q2)=x2+y2.