Solve y^4 = y^2 - x^2

Question

Solve the equation

Solve for x

Solve for y

x = ∣ y ∣ \times 1 - y^{2} x = - ∣ y ∣ \times 1 - 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

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{x}{2 y ^{3} - y}

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Find the second derivative

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y ^{6} - 4 y ^{4} + y ^{2} + 6 x ^{2} y ^{2} - x ^{2}}{8 y ^{9} - 12 y ^{7} + 6 y ^{5} - y ^{3}}

Calculate

y^{4} = y^{2} - x^{2}

Take the derivative of both sides

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

Calculate the derivative

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

Calculate the derivative

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

Move the variable to the left side

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

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

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

Calculate the derivative

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

Any expression multiplied by 1 remains the same

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

Calculate

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

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

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

Use equation \frac{d y}{d x} = - \frac{x}{2 y ^{3} - y} to substitute

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

Solution

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Calculate

- \frac{2 y ^{3} - y - 6 x y ^{2} ( - \frac{x}{2 y ^{3} - y} ) + x ( - \frac{x}{2 y ^{3} - y} )}{( 2 y ^{3} - y ) ^{2}}

Multiply

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

Multiply the terms

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

Calculate the sum or difference

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

Divide the terms

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

Expand the expression

More Steps

- \frac{4 y ^{6} - 4 y ^{4} + y ^{2} + 6 x ^{2} y ^{2} - x ^{2}}{8 y ^{9} - 12 y ^{7} + 6 y ^{5} - y ^{3}}

\frac{d ^{2} y}{d x ^{2}} = - \frac{4 y ^{6} - 4 y ^{4} + y ^{2} + 6 x ^{2} y ^{2} - x ^{2}}{8 y ^{9} - 12 y ^{7} + 6 y ^{5} - y ^{3}}

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