2000

1) Solve: pq=xmynz2l

[12M]

2) Prove that if x31+x32+x33=1 when z=0, the solution of the equation (S−x1)p1+(S−x2)p2+(S−x3)p3=S−z can be given in the form S3{(x1−z)3+(x2−z)3+(x3−z)3}4=(x1+x2+x3−3z)3 where S=x1+x2+x3+z and pi=∂z∂xi,i=1,2,3

[12M]

3) Solve by Charpit’s method the equation p2x(x−1)+2pqxy+q2y(y−1)−2pxz−2qyz+z2=0

[15M]

4) Solve: (D2−DD′−2D2)z=2x+3y+e3x+4y

[15M]

5) A tightly stretched string with fixed end points x=0,x=l is initially at rest in equilibrium position. If it is set vibrating by giving each point x of it a velocity kx(l−x), obtain at time t the displacement y at a distance x from the end x=0.

[30M]

1999

1) Verify that the differential equation (y2+yz)dx+(xz+z2)dy+(y2−xy)dz=0
is integrable and find its primitive.

[20M]

2) Find the surface which intersects the surface of the system z(x+y)=c(3z+1),c=a constant,orthogonally and which passes through the circle x2+y2=1,z=1.

[20M]

3) Find the characteristics pf the equation pq=z, and determine the integral surface which passes through the parabola x=0,y=z.

[20M]

4) Use Charpit’s method to find a complete integral to p2+q2−2px−2qy+1=0.

[20M]

5) Find the solution of the equation ∂2z∂x2+∂2z∂y2=e−1cosy
which x→0 as x→∞ and has the value cosy when x=0.

[20M]

6) One end of the string (x=0) is fixed, and the point x=a is made to oscillate, so that at time t the displacement is g(t).Show that the displacement u(x,t) of the point x at the time t is given by u(x,t)=f(ct−x)−f(ct+x) where f is a function satisying the relation f(t+2a)=f(t)−g(t+ac).

[20M]

1998

1) Find the differential equation of the set of all right circular cones whose axes coincide with the Z-axis. Form the differential equation by eliminating a, b and c from Z=a(x+y)+b(x−y)+abt+c.

[20M]

2) Solve: x∂u∂x+y∂u∂y+w∂u∂z=xyz

[20M]

3) Find the integral surface of the linear partial differential equation x(y2+z)∂z∂x−y(x2+z)∂z∂y=(x2−y2)z through the straight line x+y=0,z=1.

[20M]

4) Use Charpit’s method to find complete integral of 2x[(z∂z∂y)2+1]=z)∂z∂x.

[20M]

5) Find a real function V(x,y), which reduces to zero when y=0 and satisfies the equation ∂2V∂x2+∂2V∂y2=−4π(x2+y2).

[20M]

6) Apply Jacobi’s method to find a complete integral of the equation 2∂z∂x1x1x3+3∂z∂x2x23+(∂z∂x2)2∂z∂x3=0.

[20M]

1997

1) Find the differential equation of all surfaces of revolution having z-axis as the axis of rotation.

[10M]

2) Form the differential equation by eliminating a and b from z=(x2+a)(y2+b).

[10M]

3) Find the equation of surfinces satisfying 4yzp+q+2y=0 and passing through y2+z2=1, x+z=2

[20M]

4) Solve (y+z)p+(z+x)q=x+y.

[20M]

5) Use Charpit’s method to find complete integral of z2(p2z2+q2)=1.

[20M]

6) Solve (D3X−D3Y)z=x3y3.

[20M]

7) Apply Jacobi’s method to find complete integral of p13+p22+p3=1. Here p1=∂z∂x1,p2=∂z∂x2,p3=∂z∂x3 and z is a function of x1,x2,x3

[20M]

1996

1) Find the differential equation of all spheres of radius λ having their centre in xy-plane.

[10M]

2) Form differential equation by eliminating f and g from z=f(x2−y)+g(x2+y)

[10M]

3) Solve: z2(p2+q2+1)=c2.

[15M]

4) Find the integral surface of the equation (x−y)y2p+(y−x)x2q=(x2+y2)z passing through the curve xz=a3,y=0

[15M]

5) Apply Charpit’s method to find the complete integral of z=px+a+p2+q2

[8M]

6) Solve: ∂2z∂x2+∂2z∂y2=cosmxcosny

[15M]

7) Find a surface passing through the lines z=x=0 and z−1=x−y=0 satisfying ∂2z∂x2−4∂2z∂x∂y+4∂2z∂y2=0

[15M]

1995

1) In the context of a partial differential equation of the first order in three independent variables, define and illustrate the terms: (i) the complete integral (ii) the singular integral

[15M]

2) Find the general integral of (y+z+w)∂w∂x+(z+x+w)∂w∂y+(x+y+w)∂w∂z=x+y+z

[15M]

3) Obtain the differential equation of the surfaces which are the envelopes of a one-parameter family of planes.

[15M]

4.(a) Explain in detail the Charpit’s method of solving the nonlinear partial differential equation fx,y,z,(∂z∂x,∂z∂y)=0

[10M]

4.(b) Solve ∂z∂x1∂z∂x2∂z∂x3=z3x1x2x3

[10M]

5) Solve (D3x−7DxD2y−6D3y)z=sin(x+2y)+e3x+y

[20M]