Question

pq+x(2y+1)p+(y^2+y)q-(2y+1)z=0,using charpit method, find a complete integral of the equation​

Answer 1

Answer:

I want to solve yp2=2(z+xp+yq),where p=zx,q=zy

My attempt:Let f(x,y,z,p,q)=yp2−2(z+xp+yq)

So that fx=−2p,fy=p2−2q,fz=−2,fp=2py−2x,fq=−2y

As per Charpits method:dxfp=dyfq=dzpfp+qfq=−dpfx+pfz=−dqfy+qfz

So,putting all the values and then equating second and fourth term,I get p=c/y2 and equating fourth and fifth term gives pq=p3/12+a, where a and c are constants.

Then dz=pdx+qdy .... (A)

Here I got stuck and I don't know how to solve(A) for z.

Please help me solve this problem.Any help towards this is much appreciated.

PS:I need "complete integral" of yp2=2(z+xp+yq),i.e a solution of this form:g(x,y,z,c1,c2)=0,where c1 and c2 are arbitrary constants.

pde