Solve 2x^3y=5x-y - ScanMath

Question

Solve the equation

y = \frac{5 x}{2 x ^{3} + 1}

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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

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r = 0 r = \frac{3 5 sec ^{2} ( θ ) csc ( θ ) - sec ^{3} ( θ )}{3 2}

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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 - 6 x ^{2} y}{2 x ^{3} + 1}

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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{48 x ^{4} y - 12 x y - 60 x ^{2}}{4 x ^{6} + 4 x ^{3} + 1}

Calculate

2 x^{3} y = 5 x - y

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

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

Move the expression to the left side

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

Move the expression to the right side

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

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

(2 x^{3} + 1) \frac{d y}{d x} = 5 - 6 x^{2} y

Divide both sides

\frac{( 2 x ^{3} + 1 ) \frac{d y}{d x}}{2 x ^{3} + 1} = \frac{5 - 6 x ^{2} y}{2 x ^{3} + 1}

Divide the numbers

\frac{d y}{d x} = \frac{5 - 6 x ^{2} y}{2 x ^{3} + 1}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{5 - 6 x ^{2} y}{2 x ^{3} + 1})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{5 - 6 x ^{2} y}{2 x ^{3} + 1})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d x} ( 5 - 6 x ^{2} y ) \times ( 2 x ^{3} + 1 ) - ( 5 - 6 x ^{2} y ) \times \frac{d}{d x} ( 2 x ^{3} + 1 )}{( 2 x ^{3} + 1 ) ^{2}}

Calculate the derivative

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{( - 12 x y - 6 x ^{2} \frac{d y}{d x} ) ( 2 x ^{3} + 1 ) - ( 5 - 6 x ^{2} y ) \times \frac{d}{d x} ( 2 x ^{3} + 1 )}{( 2 x ^{3} + 1 ) ^{2}}

Calculate the derivative

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\frac{d ^{2} y}{d x ^{2}} = \frac{( - 12 x y - 6 x ^{2} \frac{d y}{d x} ) ( 2 x ^{3} + 1 ) - ( 5 - 6 x ^{2} y ) \times 6 x ^{2}}{( 2 x ^{3} + 1 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 24 x ^{4} y - 12 x y - 12 x ^{5} \frac{d y}{d x} - 6 x ^{2} \frac{d y}{d x} - ( 5 - 6 x ^{2} y ) \times 6 x ^{2}}{( 2 x ^{3} + 1 ) ^{2}}

Calculate

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\frac{d ^{2} y}{d x ^{2}} = \frac{- 24 x ^{4} y - 12 x y - 12 x ^{5} \frac{d y}{d x} - 6 x ^{2} \frac{d y}{d x} - ( 30 x ^{2} - 36 x ^{4} y )}{( 2 x ^{3} + 1 ) ^{2}}

Calculate

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{12 x ^{4} y - 12 x y - 12 x ^{5} \frac{d y}{d x} - 6 x ^{2} \frac{d y}{d x} - 30 x ^{2}}{( 2 x ^{3} + 1 ) ^{2}}

Use equation \frac{d y}{d x} = \frac{5 - 6 x ^{2} y}{2 x ^{3} + 1} to substitute

\frac{d ^{2} y}{d x ^{2}} = \frac{12 x ^{4} y - 12 x y - 12 x ^{5} \times \frac{5 - 6 x ^{2} y}{2 x ^{3} + 1} - 6 x ^{2} \times \frac{5 - 6 x ^{2} y}{2 x ^{3} + 1} - 30 x ^{2}}{( 2 x ^{3} + 1 ) ^{2}}

Solution

More Steps

\frac{d ^{2} y}{d x ^{2}} = \frac{48 x ^{4} y - 12 x y - 60 x ^{2}}{4 x ^{6} + 4 x ^{3} + 1}

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