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

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 first derivative

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{1}{2}} + \frac{d y}{d x}}{2 y ^{\frac{1}{2}}}

Multiply both sides of the equation by LCD

(2 x + 2 y \frac{d y}{d x}) \times 2 y^{\frac{1}{2}} = \frac{6 y ^{\frac{1}{2}} + \frac{d y}{d x}}{2 y ^{\frac{1}{2}}} \times 2 y^{\frac{1}{2}}

Simplify the equation

More Steps

4 x y^{\frac{1}{2}} + 4 y^{\frac{3}{2}} \frac{d y}{d x} = \frac{6 y ^{\frac{1}{2}} + \frac{d y}{d x}}{2 y ^{\frac{1}{2}}} \times 2 y^{\frac{1}{2}}

Simplify the equation

4 x y^{\frac{1}{2}} + 4 y^{\frac{3}{2}} \frac{d y}{d x} = 6 y^{\frac{1}{2}} + \frac{d y}{d x}

Move the variable to the left side

4 x y^{\frac{1}{2}} + 4 y^{\frac{3}{2}} \frac{d y}{d x} - \frac{d y}{d x} = 6 y^{\frac{1}{2}}

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

4 x y^{\frac{1}{2}} + (4 y^{\frac{3}{2}} - 1) \frac{d y}{d x} = 6 y^{\frac{1}{2}}

Move the constant to the right side

(4 y^{\frac{3}{2}} - 1) \frac{d y}{d x} = 6 y^{\frac{1}{2}} - 4 x y^{\frac{1}{2}}

Multiply both sides of the equation by \frac{1}{4 y ^{\frac{3}{2}} - 1}

(4 y^{\frac{3}{2}} - 1) \frac{d y}{d x} \times \frac{1}{4 y ^{\frac{3}{2}} - 1} = (6 y^{\frac{1}{2}} - 4 x y^{\frac{1}{2}}) \times \frac{1}{4 y ^{\frac{3}{2}} - 1}

Calculate the product

(4 y^{\frac{3}{2}} - 1) \frac{d y}{d x} \times \frac{1}{4 y ^{\frac{3}{2}} - 1} = \frac{6 y ^{\frac{1}{2}} - 4 x y ^{\frac{1}{2}}}{4 y ^{\frac{3}{2}} - 1}

Cancel out the greatest common factor 4 y^{\frac{3}{2}} - 1

\frac{d y}{d x} = \frac{6 y ^{\frac{1}{2}} - 4 x y ^{\frac{1}{2}}}{4 y ^{\frac{3}{2}} - 1}

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

More Steps

\frac{d y}{d x} = \frac{6 y - 4 x y}{4 y ^{\frac{3}{2}} - 1}

Solution

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

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