Solve ( x^2 + y^2 - 1 )*x^2 y^3 = 0

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = - y^{2} + 1 x = - - y^{2} + 1

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Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

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r = 0 r = 1 r = - 1

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{- 4 x ^{2} y - 2 y ^{3} + 2 y}{5 x y ^{2} + 3 x ^{3} - 3 x}

Calculate

(x^{2} + y^{2} - 1) \cdot x^{2} y^{3} = 0

Simplify the expression

x^{2} (x^{2} + y^{2} - 1) y^{3} = 0

Take the derivative of both sides

\frac{d}{d x} (x^{2} (x^{2} + y^{2} - 1) y^{3}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

2 x^{3} y^{3} + 2 x y^{5} - 2 x y^{3} + 2 x^{3} y^{3} + 2 x^{2} y^{4} \frac{d y}{d x} + 3 x^{4} y^{2} \frac{d y}{d x} + 3 x^{2} y^{4} \frac{d y}{d x} - 3 x^{2} y^{2} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

2 x^{3} y^{3} + 2 x y^{5} - 2 x y^{3} + 2 x^{3} y^{3} + 2 x^{2} y^{4} \frac{d y}{d x} + 3 x^{4} y^{2} \frac{d y}{d x} + 3 x^{2} y^{4} \frac{d y}{d x} - 3 x^{2} y^{2} \frac{d y}{d x} = 0

Calculate the sum or difference

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4 x^{3} y^{3} + 2 x y^{5} - 2 x y^{3} + (5 x^{2} y^{4} + 3 x^{4} y^{2} - 3 x^{2} y^{2}) \frac{d y}{d x} = 0

Move the constant to the right side

(5 x^{2} y^{4} + 3 x^{4} y^{2} - 3 x^{2} y^{2}) \frac{d y}{d x} = 0 - (4 x^{3} y^{3} + 2 x y^{5} - 2 x y^{3})

Subtract the terms

More Steps

(5 x^{2} y^{4} + 3 x^{4} y^{2} - 3 x^{2} y^{2}) \frac{d y}{d x} = - 4 x^{3} y^{3} - 2 x y^{5} + 2 x y^{3}

Divide both sides

\frac{( 5 x ^{2} y ^{4} + 3 x ^{4} y ^{2} - 3 x ^{2} y ^{2} ) \frac{d y}{d x}}{5 x ^{2} y ^{4} + 3 x ^{4} y ^{2} - 3 x ^{2} y ^{2}} = \frac{- 4 x ^{3} y ^{3} - 2 x y ^{5} + 2 x y ^{3}}{5 x ^{2} y ^{4} + 3 x ^{4} y ^{2} - 3 x ^{2} y ^{2}}

Divide the numbers

\frac{d y}{d x} = \frac{- 4 x ^{3} y ^{3} - 2 x y ^{5} + 2 x y ^{3}}{5 x ^{2} y ^{4} + 3 x ^{4} y ^{2} - 3 x ^{2} y ^{2}}

Solution

More Steps

\frac{d y}{d x} = \frac{- 4 x ^{2} y - 2 y ^{3} + 2 y}{5 x y ^{2} + 3 x ^{3} - 3 x}

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