Question #316113

Form the PDE of the following by eliminating arbitrary functions

phi(z^2-xy,x/z)


1
Expert's answer
2022-03-23T15:33:03-0400

Φ(z2xy,x/z)=0\Phi(z^2-xy,x/z)=0 (1)

We set u=z2xyu=z^2-xy; v=xzv=\frac xz.

Then the operator d on (1) leads to:

dΦ(u,v)=Φ(u,v)udu+Φ(u,v)vdv\displaystyle d\Phi(u,v)=\frac{\partial \Phi(u,v)}{\partial u}du+\frac{\partial \Phi(u,v)}{\partial v}dv=0.

Thus

0=du=d(z2xy)0=du=d(z^2-xy) (3)

0=dv=d(xz)0=dv=d(\frac{x}{z}) (4)

From (3) and (4) we have:

2zdzydxxdy=02zdz-ydx-xdy=0

1zdxxz2dz=0\displaystyle \frac1zdx-\frac{x}{z^2}dz=0

Thus getting rid of dx we obtain:

dx=xzdz\displaystyle dx=\frac xzdz

(2zyxz)dz=xdy\displaystyle (2z-y\frac xz)dz=xdy

dz=dy2z2yxxz=dxx2xz\displaystyle dz=\frac {dy}{\frac{2z^2-yx}{xz}}=\frac{dx}{\frac{x^2}{xz}}

dzxz=dy2z2yx=dxx2\displaystyle \frac{dz}{xz}=\frac {dy}{2z^2-yx}=\frac{dx}{x^2}

Finally we obtain the desired PDE for z(x,y)z(x,y):

x2zx+(2z2xy)zy=xz\displaystyle x^2\frac{\partial z}{\partial x}+(2z^2-xy)\frac{\partial z}{\partial y}=xz

Answer: x2zx+(2z2xy)zy=xz\displaystyle x^2\frac{\partial z}{\partial x}+(2z^2-xy)\frac{\partial z}{\partial y}=xz.


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