Solve x=y*x - ScanMath

Solve the equation

Solve for x

Solve for y

x = 0

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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 = 0 r = csc (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{1 - y}{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 + 2 y}{x ^{2}}

Calculate

x = y \cdot x

Take the derivative of both sides

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

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

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

Calculate the derivative

More Steps

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

Swap the sides of the equation

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

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

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

Divide both sides

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

Use differentiation rules

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

Calculate the derivative

More Steps

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

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

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

Use the commutative property to reorder the terms

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

Any expression multiplied by 1 remains the same

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

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

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

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

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

Solution

More Steps

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

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