## Problem 06 | Elimination of Arbitrary Constants

**Problem 6**

Eliminate the c_{1} and c_{2} from x = c_{1} cos ωt + c_{2} sin ωt. ω being a parameter not to be eliminated.

**Problem 6**

Eliminate the c_{1} and c_{2} from x = c_{1} cos ωt + c_{2} sin ωt. ω being a parameter not to be eliminated.

**Problem 5**

Eliminate A and B from x = A sin (ωt + B). ω being a parameter not to be eliminated.

**Problem 02**

$y \sin x - xy^2 = c$

**Solution 02**

$y \sin x - xy^2 = c$

$(y \cos x~dx + \sin x~dy) - (2xy~dy + y^2~dx) = 0$

**Problem 01**

$x^3 - 3x^2y = c$

**Solution 01**

**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)

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