We rewrite this equation in the form
z−p2x−q2y=0
which allows us to highlight the function F(x,y,z,p,q)=z−p2x−q2y , which we need for Charpit's Method.
Charpit's equation is:
Fpdx=Fqdy=pFp+qFqdz=−(Fx+pFz)dp=−(Fy+qFz)dq
In our case,
F(x,y,z,p,q)=z−p2x−q2y⟶Fp=−2pxFq=−2qypFp+qFq=−2p2x−2q2x=−2(p2x+q2y)≡−2zFx+pFz=−p2+pFy+qFz=−q2+q
Then, our equation takes the form
−2pxdx=−2qydy=−2zdz=p2−pdp=q2−qdq
1) We are interested in the first and fourth element:
−2pxdx=p2−pdp⟶−2pxdx=p(p−1)dp⟶−2xdx=(p−1)dp−21⋅∫xdx=∫p−1dp⟶−21ln∣x∣=ln∣p−1∣−ln∣A∣⟶ln∣∣xA∣∣=ln∣(p−1)∣⟶xA=p−1⟶p=xA+x
2) We are interested in the second and fifth element:
−2qydy=q2−qdq⟶−2qydy=q(q−1)dq⟶−2ydy=(q−1)dq−21⋅∫ydy=∫q−1dq⟶−21ln∣y∣=ln∣q−1∣−ln∣B∣⟶ln∣∣yB∣∣=ln∣(q−1)∣⟶yB=q−1⟶q=yB+y
It remains to substitute the obtained results in the initial equation:
z=p2x+q2y⟶z=(xA+x)2x+(yB+y)2yz=xA2+2Ax+x⋅x+yB2+2By+y⋅yz=x+y+2(Ax+By)+A2+B2
ANSWER
z=x+y+2(Ax+By)+A2+B2
Comments