We rewrite this equation in the form
z − p 2 x − q 2 y = 0 z-p^2x-q^2y=0 z − p 2 x − q 2 y = 0
which allows us to highlight the function F ( x , y , z , p , q ) = z − p 2 x − q 2 y F(x,y,z,p,q)=z-p^2x-q^2y F ( x , y , z , p , q ) = z − p 2 x − q 2 y , which we need for Charpit's Method.
Charpit's equation is:
d x F p = d y F q = d z p F p + q F q = d p − ( F x + p F z ) = d q − ( F y + q F z ) \frac{dx}{F_p}=\frac{dy}{F_q}=\frac{dz}{pF_p+qF_q}=\frac{dp}{-\left(F_x+pF_z\right)}=
\frac{dq}{-\left(F_y+qF_z\right)} F p d x = F q d y = p F p + q F q d z = − ( F x + p F z ) d p = − ( F y + q F z ) d q
In our case,
F ( x , y , z , p , q ) = z − p 2 x − q 2 y ⟶ F p = − 2 p x F q = − 2 q y p F p + q F q = − 2 p 2 x − 2 q 2 x = − 2 ( p 2 x + q 2 y ) ≡ − 2 z F x + p F z = − p 2 + p F y + q F z = − q 2 + q F(x,y,z,p,q)=z-p^2x-q^2y\longrightarrow\\[0.3cm]
F_p=-2px\quad F_q=-2qy\\[0.3cm]
pF_p+qF_q=-2p^2x-2q^2x=-2\left(p^2x+q^2y\right)\equiv-2z\\[0.3cm]
F_x+pF_z=-p^2+p\quad F_y+qF_z=-q^2+q F ( x , y , z , p , q ) = z − p 2 x − q 2 y ⟶ F p = − 2 p x F q = − 2 q y p F p + q F q = − 2 p 2 x − 2 q 2 x = − 2 ( p 2 x + q 2 y ) ≡ − 2 z F x + p F z = − p 2 + p F y + q F z = − q 2 + q
Then, our equation takes the form
d x − 2 p x = d y − 2 q y = d z − 2 z = d p p 2 − p = d q q 2 − q \frac{dx}{-2px}=\frac{dy}{-2qy}=\frac{dz}{-2z}=\frac{dp}{p^2-p}=
\frac{dq}{q^2-q} − 2 p x d x = − 2 q y d y = − 2 z d z = p 2 − p d p = q 2 − q d q
1) We are interested in the first and fourth element:
d x − 2 p x = d p p 2 − p ⟶ d x − 2 p x = d p p ( p − 1 ) ⟶ d x − 2 x = d p ( p − 1 ) − 1 2 ⋅ ∫ d x x = ∫ d p p − 1 ⟶ − 1 2 ln ∣ x ∣ = ln ∣ p − 1 ∣ − ln ∣ A ∣ ⟶ ln ∣ A x ∣ = ln ∣ ( p − 1 ) ∣ ⟶ A x = p − 1 ⟶ p = A + x x \frac{dx}{-2px}=\frac{dp}{p^2-p}\longrightarrow\frac{dx}{-2px}=\frac{dp}{p(p-1)}\longrightarrow\frac{dx}{-2x}=\frac{dp}{(p-1)}\\[0.3cm]
-\frac{1}{2}\cdot\int\frac{dx}{x}=\int\frac{dp}{p-1}\longrightarrow\\[0.3cm]
-\frac{1}{2}\ln|x|=\ln|p-1|-\ln|A|\longrightarrow
\ln\left|\frac{A}{\sqrt{x}}\right|=\ln|(p-1)|\longrightarrow\\[0.3cm]
\frac{A}{\sqrt{x}}=p-1\longrightarrow\boxed{p=\frac{A+\sqrt{x}}{\sqrt{x}}} − 2 p x d x = p 2 − p d p ⟶ − 2 p x d x = p ( p − 1 ) d p ⟶ − 2 x d x = ( p − 1 ) d p − 2 1 ⋅ ∫ x d x = ∫ p − 1 d p ⟶ − 2 1 ln ∣ x ∣ = ln ∣ p − 1∣ − ln ∣ A ∣ ⟶ ln ∣ ∣ x A ∣ ∣ = ln ∣ ( p − 1 ) ∣ ⟶ x A = p − 1 ⟶ p = x A + x
2) We are interested in the second and fifth element:
d y − 2 q y = d q q 2 − q ⟶ d y − 2 q y = d q q ( q − 1 ) ⟶ d y − 2 y = d q ( q − 1 ) − 1 2 ⋅ ∫ d y y = ∫ d q q − 1 ⟶ − 1 2 ln ∣ y ∣ = ln ∣ q − 1 ∣ − ln ∣ B ∣ ⟶ ln ∣ B y ∣ = ln ∣ ( q − 1 ) ∣ ⟶ B y = q − 1 ⟶ q = B + y y \frac{dy}{-2qy}=\frac{dq}{q^2-q}\longrightarrow\frac{dy}{-2qy}=\frac{dq}{q(q-1)}\longrightarrow\frac{dy}{-2y}=\frac{dq}{(q-1)}\\[0.3cm]
-\frac{1}{2}\cdot\int\frac{dy}{y}=\int\frac{dq}{q-1}\longrightarrow\\[0.3cm]
-\frac{1}{2}\ln|y|=\ln|q-1|-\ln|B|\longrightarrow
\ln\left|\frac{B}{\sqrt{y}}\right|=\ln|(q-1)|\longrightarrow\\[0.3cm]
\frac{B}{\sqrt{y}}=q-1\longrightarrow\boxed{q=\frac{B+\sqrt{y}}{\sqrt{y}}} − 2 q y d y = q 2 − q d q ⟶ − 2 q y d y = q ( q − 1 ) d q ⟶ − 2 y d y = ( q − 1 ) d q − 2 1 ⋅ ∫ y d y = ∫ q − 1 d q ⟶ − 2 1 ln ∣ y ∣ = ln ∣ q − 1∣ − ln ∣ B ∣ ⟶ ln ∣ ∣ y B ∣ ∣ = ln ∣ ( q − 1 ) ∣ ⟶ y B = q − 1 ⟶ q = y B + y
It remains to substitute the obtained results in the initial equation:
z = p 2 x + q 2 y ⟶ z = ( A + x x ) 2 x + ( B + y y ) 2 y z = A 2 + 2 A x + x x ⋅ x + B 2 + 2 B y + y y ⋅ y z = x + y + 2 ( A x + B y ) + A 2 + B 2 z=p^2x+q^2y\longrightarrow z=\left(\frac{A+\sqrt{x}}{\sqrt{x}}\right)^2x+\left(\frac{B+\sqrt{y}}{\sqrt{y}}\right)^2y\\[0.3cm]
z=\frac{A^2+2A\sqrt{x}+x}{x}\cdot x+\frac{B^2+2B\sqrt{y}+y}{y}\cdot y\\[0.3cm]
\boxed{z=x+y+2\left(A\sqrt{x}+B\sqrt{y}\right)+A^2+B^2} z = p 2 x + q 2 y ⟶ z = ( x A + x ) 2 x + ( y B + y ) 2 y z = x A 2 + 2 A x + x ⋅ x + y B 2 + 2 B y + y ⋅ y z = x + y + 2 ( A x + B y ) + A 2 + B 2
ANSWER
z = x + y + 2 ( A x + B y ) + A 2 + B 2 z=x+y+2\left(A\sqrt{x}+B\sqrt{y}\right)+A^2+B^2 z = x + y + 2 ( A x + B y ) + A 2 + B 2
Comments