Solution;
z(x+2y)p−z(2x+y)q=y2−x2
The auxiliary equation are;
z(x+2y)dx=−z(2x+y)dy=y2−x2dz
Considering the first two equations;
3z(x+y)dx−dy=0 .....(1)
dx=dy
Integrating;
x=y+c1
Considering (1) and the last auxillary equation;
3z(x+y)dx−dy=x2−y2dz
(x−y)(dx−dy)=3zdz
xdx−xdy−ydx+ydy=3zdz
Integrating;
2x2−2xy+2y2=23z2+c2
x2−4xy+y2=3z2+c2
The solution of the equation is;
F(x−y,x2+y2−3z2−4xy)=0
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