Solve x^2+y^2=3x+\sqrt{y}

Question

Solve the equation

x = \frac{3 + 9 - 4 y ^{2} + 4 y}{2} x = \frac{3 - 9 - 4 y ^{2} + 4 y}{2}

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Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{6 y - 4 x y}{4 y y - 1}

Calculate

x^{2} + y^{2} = 3 x + y

Take the derivative of both sides

\frac{d}{d x} (x^{2} + y^{2}) = \frac{d}{d x} (3 x + y)

Calculate the derivative

More Steps

2 x + 2 y \frac{d y}{d x} = \frac{d}{d x} (3 x + y)

Calculate the derivative

More Steps

2 x + 2 y \frac{d y}{d x} = \frac{6 y + \frac{d y}{d x}}{2 y}

Multiply both sides of the equation by LCD

(2 x + 2 y \frac{d y}{d x}) \times 2 y = \frac{6 y + \frac{d y}{d x}}{2 y} \times 2 y

Simplify the equation

More Steps

4 x y + 4 y y \times \frac{d y}{d x} = \frac{6 y + \frac{d y}{d x}}{2 y} \times 2 y

Simplify the equation

4 x y + 4 y y \times \frac{d y}{d x} = 6 y + \frac{d y}{d x}

Move the expression to the left side

4 x y + 4 y y \times \frac{d y}{d x} - \frac{d y}{d x} = 6 y

Move the expression to the right side

4 y y \times \frac{d y}{d x} - \frac{d y}{d x} = 6 y - 4 x y

Collect like terms by calculating the sum or difference of their coefficients

(4 y y - 1) \frac{d y}{d x} = 6 y - 4 x y

Divide both sides

\frac{( 4 y y - 1 ) \frac{d y}{d x}}{4 y y - 1} = \frac{6 y - 4 x y}{4 y y - 1}

Solution

\frac{d y}{d x} = \frac{6 y - 4 x y}{4 y y - 1}

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