Solve x^2 y^2=2x - ScanMath

Question

Solve the equation

Solve for x

Solve for y

x = 0 x = \frac{2}{y ^{2}}

Show Solution

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

Show Solution

r = 0 r = 3 2 sec (θ) csc^{2} (θ)

Show Solution

Find the derivative with respect to x

Find the derivative with respect to y

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

Show Solution

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 ^{4} x ^{2} - 2 y ^{2} x - 1}{x ^{4} y ^{3}}

Calculate

x^{2} y^{2} = 2 x

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

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

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

Divide both sides

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

Divide the numbers

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

Divide the numbers

More Steps

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

More Steps

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

Calculate

More Steps

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

Calculate

More Steps

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

Calculate

More Steps

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

Calculate

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

Use equation \frac{d y}{d x} = \frac{1 - x y ^{2}}{x ^{2} y} to substitute

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

Solution

More Steps

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

Show Solution