differential equation of the family of parabolas

To find the differential equation of the family of parabolas

$\displaystyle y = ax + bx^2$

we differentiate twice to obtain

$\displaystyle y^{\prime} = a + 2bx$

$\displaystyle y^{\prime \prime} = 2b$

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 + bx^2$ . The result is the differential equation

$\displaystyle y = xy^{\prime} - \frac{1}{2}x^2y^{\prime \prime}$

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

$\displaystyle xa + x^2b + (-y)1 = 0$

$\displaystyle a + 2xb + (-y^{\prime})1 = 0$

$\displaystyle 2b +(-y^{\prime \prime})1 = 0$

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.

$\displaystyle \left\vert \begin{array}{ccc}
x & x^2 & -y \\
1 & 2x & -y^{\prime} \\
0 & 2 & -y^{\prime \prime} \end{array} \right\vert = 0
$

Expansion about the third column yields the result above.

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].



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by bloftin.