Solve 4x^29y^28x^36y^4=0

Question

Solve the equation

Solve for x

Solve for y

x = 0

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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 θ = \frac{kπ}{2}, k \in Z

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Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - \frac{5 y}{6 x}

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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{55 y}{36 x ^{2}}

Calculate

4 x^{2} 9 y^{2} 8 x^{3} 6 y^{4} = 0

Simplify the expression

1728 x^{5} y^{6} = 0

Take the derivative of both sides

\frac{d}{d x} (1728 x^{5} y^{6}) = \frac{d}{d x} (0)

Calculate the derivative

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8640 x^{4} y^{6} + 10368 x^{5} y^{5} \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

8640 x^{4} y^{6} + 10368 x^{5} y^{5} \frac{d y}{d x} = 0

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

10368 x^{5} y^{5} \frac{d y}{d x} = 0 - 8640 x^{4} y^{6}

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

10368 x^{5} y^{5} \frac{d y}{d x} = - 8640 x^{4} y^{6}

Divide both sides

\frac{10368 x ^{5} y ^{5} \frac{d y}{d x}}{10368 x ^{5} y ^{5}} = \frac{- 8640 x ^{4} y ^{6}}{10368 x ^{5} y ^{5}}

Divide the numbers

\frac{d y}{d x} = \frac{- 8640 x ^{4} y ^{6}}{10368 x ^{5} y ^{5}}

Divide the numbers

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\frac{d y}{d x} = - \frac{5 y}{6 x}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- \frac{5 y}{6 x})

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Calculate the derivative

More Steps

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

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{30 \frac{d y}{d x} \times x - 5 y \times 6}{( 6 x ) ^{2}}

Calculate

\frac{d ^{2} y}{d x ^{2}} = - \frac{30 \frac{d y}{d x} \times x - 30 y}{( 6 x ) ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{30 x \frac{d y}{d x} - 30 y}{( 6 x ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = - \frac{30 x \frac{d y}{d x} - 30 y}{36 x ^{2}}

Calculate

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

Use equation \frac{d y}{d x} = - \frac{5 y}{6 x} to substitute

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

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{55 y}{36 x ^{2}}

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