ES208 โ DIFFERENTIAL EQUATIONS
Engr. Dennis E. Ganas
Mechanical Engineering Department
MECHANICAL ENGINEERING
LECTURE 2
ELIMINATION OF ARBITRARY CONSTANTS
Properties
๏ท The order of differential equation is equal to the number of arbitrary constants in the
given relation. (ex. 3 constants = 3rd order)
๏ท The differential equation is consistent with the relation.
๏ท The differential equation is free from arbitrary constants.
Examples:
1. ๐ฅ3โ3๐ฅ2๐ฆ= ๐
Differentiating will result to
3๐ฅ2 ๐๐ฅ โ3(2๐ฅ๐ฆ ๐๐ฅ + ๐ฅ2 ๐๐ฆ)= 0
3๐ฅ2 ๐๐ฅ โ 6๐ฅ๐ฆ ๐๐ฅ โ 3๐ฅ2 ๐๐ฆ = 0
Divide by 3x
๐ฅ ๐๐ฅ โ 2๐ฆ ๐๐ฅ โ ๐ฅ ๐๐ฆ = 0
(๐ โ ๐๐) ๐
๐ โ ๐ ๐
๐ =๐
2. ๐ฆ๐ ๐๐๐ฅ โ๐ฅ๐ฆ2= ๐
Differentiating we obtain
(๐ฆ๐๐๐ ๐ฅ ๐๐ฅ + ๐ ๐๐๐ฅ ๐๐ฆ) โ(2๐ฅ๐ฆ ๐๐ฆ + ๐ฆ2 ๐๐ฅ) = 0
๐ฆ๐๐๐ ๐ฅ ๐๐ฅ + ๐ ๐๐๐ฅ ๐๐ฆ โ 2๐ฅ๐ฆ ๐๐ฆ โ ๐ฆ2 ๐๐ฅ = 0
(๐ฆ๐๐๐ ๐ฅ ๐๐ฅ โ ๐ฆ2 ๐๐ฅ) + (๐ ๐๐๐ฅ ๐๐ฆ โ 2๐ฅ๐ฆ ๐๐ฆ) = 0
๐(๐๐๐๐ โ ๐) ๐
๐ + (๐๐๐๐ โ ๐๐๐) ๐
๐ = ๐
