Solve 4x^26xy -10y^2 = 0

Question

Solve the equation

Solve for x

Solve for y

x = \frac{3 90 y}{6}

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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 = \frac{15 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{6} r = - \frac{15 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{6}

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{18 x ^{2} y}{6 x ^{3} - 5 y}

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Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = \frac{2592 x ^{7} y + 540 y ^{2} x ^{4} - 900 y ^{3} x}{216 x ^{9} - 540 x ^{6} y + 450 x ^{3} y ^{2} - 125 y ^{3}}

Calculate

4 x^{2} 6 x y - 10 y^{2} = 0

Simplify the expression

24 x^{3} y - 10 y^{2} = 0

Take the derivative of both sides

\frac{d}{d x} (24 x^{3} y - 10 y^{2}) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

72 x^{2} y + 24 x^{3} \frac{d y}{d x} - 20 y \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

72 x^{2} y + 24 x^{3} \frac{d y}{d x} - 20 y \frac{d y}{d x} = 0

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

72 x^{2} y + (24 x^{3} - 20 y) \frac{d y}{d x} = 0

Move the constant to the right side

(24 x^{3} - 20 y) \frac{d y}{d x} = 0 - 72 x^{2} y

Removing 0 doesn't change the value,so remove it from the expression

(24 x^{3} - 20 y) \frac{d y}{d x} = - 72 x^{2} y

Divide both sides

\frac{( 24 x ^{3} - 20 y ) \frac{d y}{d x}}{24 x ^{3} - 20 y} = \frac{- 72 x ^{2} y}{24 x ^{3} - 20 y}

Divide the numbers

\frac{d y}{d x} = \frac{- 72 x ^{2} y}{24 x ^{3} - 20 y}

Divide the numbers

More Steps

\frac{d y}{d x} = - \frac{18 x ^{2} y}{6 x ^{3} - 5 y}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{18 x ^{2} y}{6 x ^{3} - 5 y})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{18 x ^{2} y}{6 x ^{3} - 5 y})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( 18 x ^{2} y ) \times ( 6 x ^{3} - 5 y ) - 18 x ^{2} y \times \frac{d}{d x} ( 6 x ^{3} - 5 y )}{( 6 x ^{3} - 5 y ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{( 36 x y + 18 x ^{2} \frac{d y}{d x} ) ( 6 x ^{3} - 5 y ) - 18 x ^{2} y \times \frac{d}{d x} ( 6 x ^{3} - 5 y )}{( 6 x ^{3} - 5 y ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{( 36 x y + 18 x ^{2} \frac{d y}{d x} ) ( 6 x ^{3} - 5 y ) - 18 x ^{2} y ( 18 x ^{2} - 5 \frac{d y}{d x} )}{( 6 x ^{3} - 5 y ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{216 x ^{4} y - 180 x y ^{2} + 108 x ^{5} \frac{d y}{d x} - 90 x ^{2} y \frac{d y}{d x} - 18 x ^{2} y ( 18 x ^{2} - 5 \frac{d y}{d x} )}{( 6 x ^{3} - 5 y ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{216 x ^{4} y - 180 x y ^{2} + 108 x ^{5} \frac{d y}{d x} - 90 x ^{2} y \frac{d y}{d x} - ( 324 x ^{4} y - 90 x ^{2} y \frac{d y}{d x} )}{( 6 x ^{3} - 5 y ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{- 108 x ^{4} y - 180 x y ^{2} + 108 x ^{5} \frac{d y}{d x}}{( 6 x ^{3} - 5 y ) ^{2}}

Use equation \frac{d y}{d x} = - \frac{18 x ^{2} y}{6 x ^{3} - 5 y} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{- 108 x ^{4} y - 180 x y ^{2} + 108 x ^{5} ( - \frac{18 x ^{2} y}{6 x ^{3} - 5 y} )}{( 6 x ^{3} - 5 y ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{2592 x ^{7} y + 540 y ^{2} x ^{4} - 900 y ^{3} x}{216 x ^{9} - 540 x ^{6} y + 450 x ^{3} y ^{2} - 125 y ^{3}}

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