1988

1) Describe the Charpit’s method of solving the equation f(x,y,z,p,q)=0.

2) Solve xp2−ypq+y2q−y2z=0.

3) Solve (y2+z2)p−yxq=−zx.

4) Solve completely z=px+qy+3p13q13.

5) Solve x2p+y2q=z.

6) Solve 9(p2z+q2)=4.

1987

1) Solve

(i) x2∂2z∂x2−y2∂2z∂y2=xy
(ii) s−t=xy−2
(iii) r=a2t

2) Solve the boundary value problem ∂2y(x,t)∂t2=a2∂2y(x,t)∂x2 under the boundary conditions:

y(0,t)=0,y=(l,t)=Asinwt, ∂y∂t=0 at t=0, y(x,0)=0 at t=0.

3) Find the function u(x,y) which satisfies the Laplace’s equation in the rectangle 0<x<a,0<y<b, and which also satisfies the boundary conditions uy(x,0)=0, uy(x,b)=0, ux(0,y)=0, ux(a,y)=f(y).

1986

1) Solve (i) p2+q2−2px−2qy+1=0.
(ii) ∂2z∂x2+3∂2z∂x∂y+2∂2z∂y2=x+y.

2) Solve ∂2y∂x2=1c2∂2y∂t2 under the boundary conditions:
y=0 for x=0 and for all values of t
∂y∂t=0 for t=0 and for all values of x
y=0 for x=π and for all values of t
y=hxd for 0′′x′′d and t=0 and
y=h(l−x)(l−d) for d≤x≤π and t=0.

1985

1) Solve the differential equation

2) Reduce the equation y2∂2z∂x2−2xy∂2z∂x∂y+x2∂2z∂y2=y2y∂z∂x+x2x∂z∂y to canonical form and hence solve it.

3) Find a solution of ∂2y∂t2=c2∂2y∂x2 such that:
(i) $y$ involves a trigonometrically.
(ii) y=0 when x=0 or π, for all values of t
(iii) ∂y∂t=0 when t=0 for all values of x
(iv) y=sinx from x=0 to x=π2y=0 from x=π2 to x=π}

when t=0

TBC

1984

1) Find a complete integral of the equation 2zq2−y2p+y2q=0.

2) Solve the equation ∂2z∂x2−3∂2z∂x∂y+2∂2z∂y2−∂z∂x+2∂z∂y=(2+4x)e−y.

3) Solve xq2r−2xpqs+xp2t+2pq2=0.

4) Solve Laplace’s equation ρ2∂2y∂ρ2+ρ∂y∂ρ+∂2y∂θ2=0 under the boundary condition. y(ρ,0)=0,0≤ρ<10
y(ρ,π)=0,0<ρ<10 y(10,θ)=200θπ,0≤θ<π2
y(10,θ)=200π(π−θ),π2<θ<π

1983

1) Find the complete and singular integral of the differential equation z=xp+yq+p2−q2, 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 ∂2y∂t2=a2∂2y∂x2 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−x2),0<x<L and (∂y∂t)t=0=0 0<x<L where m is a suitable constant.

4) Find the solution of the heat equation ∂u∂t=c2∂2u∂x2 in the case of a semi-infinite bar extending from 0 to ∞, 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)=1√π[∫∞−π/τf(x+τw)e−w2dw−∫∞x/τf(−x+τw)e−w2dw],

where τ=2c√t