12. Form the partial differential equation of all planes through the
origin.
Ans: z=px+qy
8.5 FORMATION OF PARTIAL DIFFERENTIAL
EQUATIONS BY ELIMINATION OF ARBITRARY
FUNCTIONS
(a) Elimination of one arbitrary function of the form z=f(u)
where u=u(x, y, z)
Let
z=f (u) (8.45)
where f (u) is an arbitrary function of u where u=u(x, y, z) a known
function of x, y and z.
Differentiating (8.45) partially with respect to x and y
,
uuuu
pfpqfq
xzyz
⎛⎞
∂∂∂∂
⎛⎞
=⋅+=⋅+
′′
⎜⎟
⎜⎟
⎝⎠
∂∂∂∂
⎝⎠
(8.46), (8.47)
w
here ′ on f denotes differentiation with respect to the ar
gument u.
Eliminating f from Eqs. (8.46)–(8.47) ...
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