PlanetPhysics/Differential Equation of the Family of Parabolas
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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.
References
[edit | edit source][1] Lass, Harry. "Elements of pure and applied mathematics" New York: McGraw-Hill Companies, 1957.