IFoS PYQs 5
2004
1) Find the general solution of the partial differential equation (z2−2yz−y2)p+x(y+z)q=x(y−z)
2) Apply charpit’s method to find the complete integral of the partial differential equation pxy+pq+qy=yz.
3) Solve the Laplace equation ∂2u∂x2+∂2u∂y2=0,0≤x≤π,0≤y≤∞ satisfying the boundary conditions u(0,y)=0, for 0≤y<∞ u(π,y)=0, for 0<y<∞ u(x,∞)=0, for 0<x<π and u(x,0)=u0, for 0<x<π
2003
1) Find the general solution of the partial differential equation (mz−ny)∂z∂x+(nx−lz)∂z∂y=ly−mx.
2) Form the partial differential equation by eliminating the arbitrary function from f(x2+y2,z−xy)=0,z=z(z,y).
3) Solve ∂u∂t=c2∂2u∂t2 given that (i) u=0, When t=0t=0 for all t (ii) u=0, When x=l for all t (iii) u=x in (0,l/2)=l−x in (l/2,l)} at t=0
2002
1) Solve completely xpq=xq+yp+√pq.
2) Using charpit’s method solve completely p2−q2=(x+y)2.
3) Obtain the general solution of the following equation ∂2z∂x2−3∂2z∂x∂y+2∂2z∂y2=(2+4x)ex−2y.
2001
1) Find the complete integral of the partial differential equation x2p2+y2q2=z2.
2) Solve by Charpit’s method (p2+q2)y=qz.
3) If φ(x) is a continuous and bounded function for −∞<x<∞, prove that the function u(x,t)=12√πkt∫∞−∞φ(ξ)e−(x−ξ)2/4xdξ is a solution of the initial value problem: ut−kuxx=0,−∞<x<∞,t>0 u(x,0)=φ(x) for −∞<x<∞.
2000
1) Solve the following initial value problem (y+z)zx+yzy=x−y;z=1+t on die initial curve C:x=t,y=1;−∞<T<∞.
2) Determine the complete integral of the equation z∂z∂x∂z∂y=(∂z∂x)2(x∂z∂y+[∂z∂x]2)+(∂z∂y)2(y∂z∂x+[∂z∂y]2).
3) Solve the following partial differential equation ∂2z∂x2+∂2z∂x∂y−2∂2z∂y2−∂z∂x−2∂z∂y=0.