Question

Find the complete integral of p2+q2=x+y

Answer 1

Step-by-step explanation:

My attempt at solution

p2y=qz−q2=a...(I)

This equation is of the form f1(x,p)=f2(y,q).

Its solution is given by dz=pdx+qdy, upon integrating this we get value of z.

From (I)

−yq2+zq−a=0, solving the quadratic equation for q, we get

q=−z±z2−4ay√−2y.

Taking the positive value only, q=−z+z2−4ay√−2y .

Also, from (I), p2y=a, therefore p=a√y.

Therfore dz=a√ydx+−z+z2−4ay√−2ydy .

I can't get any further. I know, how to solve by charpit's method, but my book mentions that I solve it without using charpit.

The answer is given: z2=(cx+a)2+c2y2 .