arbitrary constants | MATHalino reviewer about arbitrary constants

Problem 6
Eliminate the c₁ and c₂ from x = c₁ cos ωt + c₂ 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

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

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

Problem 01
$x^3 - 3x^2y = c$

Solution 01

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

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 c₁ and c₂ from the relation $y = c_1 e^{-3x} + c_2 e^{2x}$.

Solution

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

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

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