To find the differential equation of the family of parabolas
we differentiate twice to obtain
The last equation is solved for
, and the result is substituted into the previous equation. This equation is solved for
, and the expressions for
and
are substituted into
. The result is the differential equation
The elimination of the constants
and
can also be obtained by considering the equations
as a system
of homogeneous linear equations in
,
,
. The solution
is nontrivial, and hence the determinant
of the coefficients vanishes.
Expansion about the third column yields the result above.
[1] Lass, Harry. "Elements of pure and applied mathematics" New York: McGraw-Hill Companies, 1957.
This entry is a derivative of the Public domain work [1].
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