Solve 2x^2-2xy y^2-1=0

Question

Solve the equation

Solve for x

Solve for y

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

Evaluate

2 x^{2} - 2 x y \times y^{2} - 1 = 0

Multiply

More Steps

2 x^{2} - 2 x y^{3} - 1 = 0

Rewrite the expression

2 x^{2} - 2 y^{3} x - 1 = 0

Substitute a= 2,b= - 2 y^{3} and c= - 1 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

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

Simplify the expression

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

Simplify the expression

More Steps

x = \frac{2 y ^{3} \pm 4 y ^{6} + 8}{4}

Simplify the radical expression

More Steps

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

Separate the equation into 2 possible cases

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

Simplify the expression

More Steps

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

Solution

More Steps

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

Show Solution

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

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

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

Show Solution

Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

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

Calculate

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

Simplify the expression

2 x^{2} - 2 x y^{3} - 1 = 0

Take the derivative of both sides

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

Calculate the derivative

More Steps

4 x - 2 y^{3} - 6 x y^{2} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

4 x - 2 y^{3} - 6 x y^{2} \frac{d y}{d x} = 0

Move the expression to the right-hand side and change its sign

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

Subtract the terms

More Steps

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

More Steps

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{6 x y ^{2} - 9 y ^{4} x \frac{d y}{d x} - ( 2 x - y ^{3} ) ( 3 y ^{2} + 6 x y \frac{d y}{d x} )}{( 3 x y ^{2} ) ^{2}}

Calculate

More Steps

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

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{- 3 y ^{4} x \frac{d y}{d x} + 3 y ^{5} - 12 x ^{2} y \frac{d y}{d x}}{( 3 x y ^{2} ) ^{2}}

Calculate

More Steps

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

Calculate

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

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

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

Solution

More Steps

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

Show Solution