# Elimination of Arbitrary Constants

**Properties**

- The order of differential equation is equal to the number of arbitrary constants in the given relation.
- The differential equation is consistent with the relation.
- The differential equation is free from arbitrary constants.

**Example**

Eliminate the arbitrary constants c_{1} and c_{2} from the relation $y = c_1 e^{-3x} + c_2 e^{2x}$.

**Solution**

$y = c_1 e^{-3x} + c_2 e^{2x}$ → equation (1)

$y' = -3c_1 e^{-3x} + 2c_2 e^{2x}$ → equation (2)

$y'' = 9c_1 e^{-3x} + 4c_2 e^{2x}$ → equation (3)

3 × equation (1) + equation (2)

$3y + y' = 5c_2 e^{2x}$ → equation (4)

3 × equation (2) + equation (3)

$3y' + y'' = 10c_2 e^{2x}$ → equation (5)

2 × equation (4) - equation (5)

$2(3y + y') - (3y' + y'') = 0$

$6y + 2y' - 3y' - y'' = 0$

$6y - y' - y'' = 0$ *answer*

Note: The methods of elimination vary with the way in which the constants enter the given relation.

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