Given a family of curves with n arbitrary constants, the DE is formed by differentiating n times and eliminating the constants. The resulting DE has order n.
Step-by-Step Method:
- Write the family equation with n constants: F(x, y, , ..., ) = 0
- Differentiate successively: get n equations involving y', y'', ..., y^(n)
- From the original + n derived equations (total n+1), eliminate , ...,
- The result is the DE
Standard Families and Their DEs:
| Family | Constants | DE |
|---|---|---|
| y = mx + c (all lines) | 2 | y'' = 0 |
| y = Ae^(kx), k fixed | 1 | y' = ky |
| y = Ae^(kx) + Be^(-kx) | 2 | y'' = y |
| y = A sin x + B cos x | 2 | y'' + y = 0 |
| + = , r fixed | 0 (center given) | x + yy' = 0 |
| (x-a)^2 + = | 1 | - + 2xyy' = 0 |
| = 4ax | 1 | 2xy' = y |
| xy = c | 1 | y + xy' = 0 |
Tricks:
- For y = + Be^(2x): instead of eliminating A, B directly, use the characteristic equation approach. y', y'' give a system; the DE is y'' - 3y' + 2y = 0 (roots 1, 2 of the characteristic equation).
- For circles with both center coordinates and radius as parameters: 3 constants, order 3 DE.
- If the family is given implicitly, implicit differentiation is needed.