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[parent] differential equation of the family of parabolas (Example)

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].



"differential equation of the family of parabolas" is owned by bloftin.

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Cross-references: work, domain, determinant, system, differential equation

This is version 1 of differential equation of the family of parabolas, born on 2010-02-15.
Object id is 841, canonical name is DifferentialEquationOfTheFamilyOfParabolas.
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Classification:
Physics Classification: 02.30.Hq (Ordinary differential equations)
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