Question #96969
by Charpits AE sobylve z=p^2x+q^2y
1
Expert's answer
2019-10-22T07:58:56-0400

We rewrite this equation in the form


zp2xq2y=0z-p^2x-q^2y=0



which allows us to highlight the function F(x,y,z,p,q)=zp2xq2yF(x,y,z,p,q)=z-p^2x-q^2y , which we need for Charpit's Method.


Charpit's equation is:



dxFp=dyFq=dzpFp+qFq=dp(Fx+pFz)=dq(Fy+qFz)\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)}

In our case,


F(x,y,z,p,q)=zp2xq2yFp=2pxFq=2qypFp+qFq=2p2x2q2x=2(p2x+q2y)2zFx+pFz=p2+pFy+qFz=q2+qF(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

Then, our equation takes the form



dx2px=dy2qy=dz2z=dpp2p=dqq2q\frac{dx}{-2px}=\frac{dy}{-2qy}=\frac{dz}{-2z}=\frac{dp}{p^2-p}= \frac{dq}{q^2-q}



1) We are interested in the first and fourth element:


dx2px=dpp2pdx2px=dpp(p1)dx2x=dp(p1)12dxx=dpp112lnx=lnp1lnAlnAx=ln(p1)Ax=p1p=A+xx\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) We are interested in the second and fifth element:



dy2qy=dqq2qdy2qy=dqq(q1)dy2y=dq(q1)12dyy=dqq112lny=lnq1lnBlnBy=ln(q1)By=q1q=B+yy\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}}}

It remains to substitute the obtained results in the initial equation:



z=p2x+q2yz=(A+xx)2x+(B+yy)2yz=A2+2Ax+xxx+B2+2By+yyyz=x+y+2(Ax+By)+A2+B2z=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}

ANSWER



z=x+y+2(Ax+By)+A2+B2z=x+y+2\left(A\sqrt{x}+B\sqrt{y}\right)+A^2+B^2


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