# Eliminate the Arbitrary Constants

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Obtain the differential equation by eliminating the arbitrary constants.
1. Cxsiny+x^(2)y=Cy
2. y=Ae^(-3x)-Be^(2x)+Cx^3
3. Ay=e^(Bx^2)

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### Obtain the differential…

Obtain the differential equation by eliminating the arbitrary constants

1. $Cx \sin y + x^2 y = Cy$

Kindly check my solution below. Do you have any answer key for each problem? It will be of great help to those who wish to answer if you post it here.
$Cx \sin y + x^2 y = Cy$

$x^2 y = C(y - x \sin y)$

$\dfrac{x^2 y}{y - x \sin y} = C$

$\dfrac{(y - x \sin y)(x^2 y' + 2xy) - x^2 y \left[ y' - (x \cos y y' + \sin y)\right]}{(y - x \sin y)^2} = 0$

$(y - x \sin y)(x^2 y' + 2xy) - x^2 y \left[ y' - (x \cos y y' + \sin y)\right] = 0$

$x^2 (y - x \sin y)y' + 2xy(y - x \sin y) - x^2 y y' + x^3 y \cos y y' + x^2 y \sin y = 0$

$\left[ x^2 (y - x \sin y) - x^2 y + x^3 y \cos y \right] y' + 2xy(y - x \sin y) + x^2 y \sin y = 0$

$(x^2 y - x^3 \sin y - x^2 y + x^3 y \cos y) y' + 2xy^2 - 2x^2y \sin y + x^2 y \sin y = 0$

$(-x^3 \sin y + x^3 y \cos y) y' + 2xy^2 - x^2 y \sin y = 0$

$x^3(y \cos y - \sin y) y' = x^2 y \sin y - 2xy^2$

$x^3(y \cos y - \sin y) y' = x(xy \sin y - 2y^2)$

$y' = \dfrac{x(xy \sin y - 2y^2)}{x^3(y \cos y - \sin y)}$

$y' = \dfrac{xy \sin y - 2y^2}{x^2(y \cos y - \sin y)}$