To find the differential equation of the family of parabolas

we differentiate twice to obtain

The last equation is solved for b, and the result is substituted into the previous equation. This equation is solved for a, and the expressions for a and b are substituted into y = ax + bx². The result is the differential equation

The elimination of the constants a and b can also be obtained by considering the equations

as a system of homogeneous linear equations in a,b,1. The solution (a,b, 1) is nontrivial, and hence the determinant of the coefficients vanishes.

Expansion about the third column yields the result above.

0.1 References

[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].