Solve y=xy-2 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

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

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Testing for symmetry

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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Rewrite the equation

r = \frac{sin ( θ ) - sin ^{2} ( θ ) + 4 sin ( 2 θ )}{sin ( 2 θ )} r = \frac{sin ( θ ) + sin ^{2} ( θ ) + 4 sin ( 2 θ )}{sin ( 2 θ )}

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

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

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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}{1 - 2 x + x ^{2}}

Calculate

y = x y - 2

Take the derivative of both sides

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

Calculate the derivative

More Steps

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

Calculate the derivative

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

Move the variable to the left side

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

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

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

Divide both sides

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

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

Calculate the derivative

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

Calculate

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

Multiplying or dividing an odd number of negative terms equals a negative

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

Calculate

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

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

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

Solution

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

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Conic

\frac{( x ^{'} - \frac{2}{2} ) ^{2}}{4} - \frac{( y ^{'} + \frac{2}{2} ) ^{2}}{4} = 1

Evaluate

y = x y - 2

Move the expression to the left side

y - (x y - 2) = 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

y - x y + 2 = 0

The coefficients A,B and C of the general equation are A= 0,B= - 1 and C= 0

A = 0 B = - 1 C = 0

To find the angle of rotation θ,substitute the values of A,B and C into the formula cot (2 θ) = \frac{A - C}{B}

cot (2 θ) = \frac{0 - 0}{- 1}

Calculate

cot (2 θ) = 0

Using the unit circle,find the smallest positive angle for which the cotangent is 0

2 θ = \frac{π}{2}

Calculate

θ = \frac{π}{4}

To rotate the axes,use the equation of rotation and substitute \frac{π}{4} for θ

x = x^{'} cos (\frac{π}{4}) - y^{'} sin (\frac{π}{4}) y = x^{'} sin (\frac{π}{4}) + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} sin (\frac{π}{4}) y = x^{'} sin (\frac{π}{4}) + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2} y = x^{'} sin (\frac{π}{4}) + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2} y = x^{'} \times \frac{2}{2} + y^{'} cos (\frac{π}{4})

Calculate

x = x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2} y = x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}

Substitute x and y into the original equation y - x y + 2 = 0

x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2} - (x^{'} \times \frac{2}{2} - y^{'} \times \frac{2}{2}) (x^{'} \times \frac{2}{2} + y^{'} \times \frac{2}{2}) + 2 = 0

Calculate

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\frac{2}{2} x^{'} + \frac{2}{2} y^{'} + \frac{1}{2} (y^{'})^{2} - \frac{1}{2} (x^{'})^{2} + 2 = 0

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

\frac{2}{2} x^{'} + \frac{2}{2} y^{'} + \frac{1}{2} (y^{'})^{2} - \frac{1}{2} (x^{'})^{2} = 0 - 2

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

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

Use the commutative property to reorder the terms

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

To complete the square, the same value needs to be subtract from both sides

- \frac{1}{2} (x^{'})^{2} + \frac{2}{2} x^{'} - \frac{1}{4} + \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} = - 2 - \frac{1}{4}

Factor out - \frac{1}{2} from the expression

- \frac{1}{2} ((x^{'})^{2} - 2 \times x^{'} + \frac{1}{2}) + \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} = - 2 - \frac{1}{4}

Use a^{2} - 2 ab + b^{2} = (a - b)^{2} to factor the expression

- \frac{1}{2} (x^{'} - \frac{2}{2})^{2} + \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} = - 2 - \frac{1}{4}

Subtract the numbers

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- \frac{1}{2} (x^{'} - \frac{2}{2})^{2} + \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} = - \frac{9}{4}

To complete the square, the same value needs to be added to both sides

- \frac{1}{2} (x^{'} - \frac{2}{2})^{2} + \frac{1}{2} (y^{'})^{2} + \frac{2}{2} y^{'} + \frac{1}{4} = - \frac{9}{4} + \frac{1}{4}

Factor out \frac{1}{2} from the expression

- \frac{1}{2} (x^{'} - \frac{2}{2})^{2} + \frac{1}{2} ((y^{'})^{2} + 2 \times y^{'} + \frac{1}{2}) = - \frac{9}{4} + \frac{1}{4}

Use a^{2} + 2 ab + b^{2} = (a + b)^{2} to factor the expression

- \frac{1}{2} (x^{'} - \frac{2}{2})^{2} + \frac{1}{2} (y^{'} + \frac{2}{2})^{2} = - \frac{9}{4} + \frac{1}{4}

Add the numbers

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

Multiply both sides of the equation by - \frac{1}{2}

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

Multiply the terms

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\frac{1}{4} (x^{'} - \frac{2}{2})^{2} - \frac{1}{4} (y^{'} + \frac{2}{2})^{2} = - 2 (- \frac{1}{2})

Multiply the terms

More Steps

\frac{1}{4} (x^{'} - \frac{2}{2})^{2} - \frac{1}{4} (y^{'} + \frac{2}{2})^{2} = 1

Use a = \frac{1}{\frac{1}{a}} to transform the expression

\frac{( x ^{'} - \frac{2}{2} ) ^{2}}{4} - \frac{1}{4} (y^{'} + \frac{2}{2})^{2} = 1

Solution

\frac{( x ^{'} - \frac{2}{2} ) ^{2}}{4} - \frac{( y ^{'} + \frac{2}{2} ) ^{2}}{4} = 1

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