Question

solve z^2=pqxy using charpit's method

Answer 1

The auxiliary equations aredxqxy=dypxy=dz2pqxy=dp2p−pqy=dq2zq−pqx. Hencedzz=d(px+qy)px+qy. Hence z=a(px+qy), a being a parameter. Hence the required one-parameter family of p.d.e. which is compatible with f=0 is g(x,y,z,p,q,a)=z-a(px+qy)=0.Solving for p and q from f=0 and g=0, we obtain p=pcxand q=czy, where a(c+1c)=1. Thus dz=(1cdxx+cdyy)z. On integrating, z=bx1/cyc. Hence F(x,y,z,b,c)=z- bx1/cyc=0 is acomplete integral of f=0.NoteIt can be easily verified that the matrix (FbFbxFbyFcFcxFcy)is of rank two.Example 3.12 Find a complete integral of f= (p2+q2)y-qz=0

Answer 2

The auxiliary equations aredxqxy=dypxy=dz2pqxy=dp2p−pqy=dq2zq−pqx. Hencedzz=d(px+qy)px+qy. Hence z=a(px+qy), a being a parameter. Hence the required one-parameter family of p.d.e. which is compatible with f=0 is g(x,y,z,p,q,a)=z-a(px+qy)=0.Solving for p and q from f=0 and g=0, we obtain p=pcxand q=czy, where a(c+1c)=1. Thus dz=(1cdxx+cdyy)z. On integrating, z=bx1/cyc. Hence F(x,y,z,b,c)=z- bx1/cyc=0 is acomplete integral of f=0.NoteIt can be easily verified that the matrix (FbFbxFbyFcFcxFcy)is of rank two.Example 3.12 Find a complete integral of f= (p2+q2)y-qz=0