(y-zx)p+(x+yz)q=x2+y2
dxy−xz=dyx+yz=dzx2+y2=tdx=yt−xzt,dy=xt+yzt,dz=(x2+y2)tydx+xdy=y2t−xyzt+x2t+xyzt=(x2+y2)t⟹ydx+xdyx2+y2=dzx2+y2=t⟹dz=ydx+xdy⟹zx′=y,zy′=x⟹z(x,y)=xy+c;check:(y−xz)p+(x+yz)q=(y−x⋅(xy+c))⋅zx′+(x+y⋅(xy+c))⋅zy′=y2−xy(xy+c)+x2+xy(xy+c)=x2+y2;\frac{dx}{y-xz}=\frac{dy}{x+yz}=\frac{dz}{x^2+y^2}=t\\ dx=yt-xzt,\quad dy=xt+yzt,\quad dz=(x^2+y^2)t\\ ydx+xdy=y^2t-xyzt+x^2t+xyzt=(x^2+y^2)t\Longrightarrow\\ \frac{ydx+xdy}{x^2+y^2}=\frac{dz}{x^2+y^2}=t\Longrightarrow\\ dz=ydx+xdy\Longrightarrow z'_x=y,\quad z'_y=x \Longrightarrow z(x,y)=xy+c;\\ check:\\ (y-xz)p+(x+yz)q=(y-x\cdot (xy+c))\cdot z'_x + (x+y\cdot (xy+c))\cdot z'_y=\\ y^2-xy(xy+c)+x^2+xy(xy+c)=x^2+y^2;y−xzdx=x+yzdy=x2+y2dz=tdx=yt−xzt,dy=xt+yzt,dz=(x2+y2)tydx+xdy=y2t−xyzt+x2t+xyzt=(x2+y2)t⟹x2+y2ydx+xdy=x2+y2dz=t⟹dz=ydx+xdy⟹zx′=y,zy′=x⟹z(x,y)=xy+c;check:(y−xz)p+(x+yz)q=(y−x⋅(xy+c))⋅zx′+(x+y⋅(xy+c))⋅zy′=y2−xy(xy+c)+x2+xy(xy+c)=x2+y2;
Need a fast expert's response?
and get a quick answer at the best price
for any assignment or question with DETAILED EXPLANATIONS!
Comments