:Here the given partial differential equation is
f(x,y,z,p,q)=p−(qy+z)2=0, −(1)
Now, the auxiliary equations are
∂x∂f+p∂z∂fdp=∂y∂f+q∂z∂fdp=−p∂p∂f−q∂q∂fdz=−∂p∂fdx=−∂q∂fdy
or,
2p(qy+z)dp=2q(qy+z)dq=(−p)(−1)−q.2(qy+z)ydz=−(−1)dx=−2y(qy+z)dy
Taking the 1st and 5th ratios, we get
pdp+ydy=0,
Integrating, logp+logy=loga , which gives, p=ya.
Substitute in (1), we have,ya=(qy+z)2 , which gives, q=ya−yz
Substituting these value of p and q in dz=pdx+qdy , we get,
dz=yadx+(ya−yz)dy
or
ydz+zdy=adx+yady
After integration, we get, yz=ax+2(ay)+b , which is a complete integral
with a and b are arbitrary constants.
Comments